TPTP Problem File: SWW319+1.p

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%------------------------------------------------------------------------------
% File     : SWW319+1 : TPTP v7.4.0. Released v5.2.0.
% Domain   : Software Verification
% Problem  : Hoare's Logic with Procedures 435026, 5000 axioms selected
% Version  : Especial.
% English  :

% Refs     : [BN10]  Boehme & Nipkow (2010), Sledgehammer: Judgement Day
%          : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% Source   : [Bla11]
% Names    : hoare_435026.5000.p [Bla11]

% Status   : Theorem
% Rating   : 0.75 v7.4.0, 0.77 v7.3.0, 0.79 v7.1.0, 0.78 v7.0.0, 0.80 v6.4.0, 0.77 v6.3.0, 0.79 v6.2.0, 0.88 v6.1.0, 0.93 v6.0.0, 0.91 v5.5.0, 0.93 v5.4.0, 0.96 v5.3.0, 1.00 v5.2.0
% Syntax   : Number of formulae    : 5227 (1675 unit)
%            Number of atoms       : 12728 (4454 equality)
%            Maximal formula depth :   22 (   6 average)
%            Number of connectives : 8494 ( 993   ~; 152   |; 719   &)
%                                         ( 904 <=>;5726  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of predicates  :  148 (   0 propositional; 1-6 arity)
%            Number of functors    :  385 (  50 constant; 0-7 arity)
%            Number of variables   : 16628 (   0 sgn;16473   !; 155   ?)
%            Maximal term depth    :   19 (   3 average)
% SPC      : FOF_THM_RFO_SEQ

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2011-03-01 11:47:21
%------------------------------------------------------------------------------
%----Relevant facts (4983)
fof(fact_ext,axiom,(
    ! [V_g_2,V_f_2] :
      ( ! [B_x] : hAPP(V_f_2,B_x) = hAPP(V_g_2,B_x)
     => V_f_2 = V_g_2 ) )).

fof(fact_empty,axiom,(
    ! [V_Ga_2,T_b] : c_Hoare__Mirabelle_Ohoare__derivs(T_b,V_Ga_2,c_Orderings_Obot__class_Obot(tc_fun(tc_Hoare__Mirabelle_Otriple(T_b),tc_HOL_Obool))) )).

fof(fact_hoare__derivs_Oequations_I1_J,axiom,(
    ! [V_a_2,T_b] : c_Hoare__Mirabelle_Ohoare__derivs(T_b,V_a_2,c_Orderings_Obot__class_Obot(tc_fun(tc_Hoare__Mirabelle_Otriple(T_b),tc_HOL_Obool))) )).

fof(fact_cut,axiom,(
    ! [V_Ga_2,V_tsa_2,V_G_H_2,T_b] :
      ( c_Hoare__Mirabelle_Ohoare__derivs(T_b,V_G_H_2,V_tsa_2)
     => ( c_Hoare__Mirabelle_Ohoare__derivs(T_b,V_Ga_2,V_G_H_2)
       => c_Hoare__Mirabelle_Ohoare__derivs(T_b,V_Ga_2,V_tsa_2) ) ) )).

fof(fact_hoare__derivs_Oinsert,axiom,(
    ! [V_tsa_2,V_ta_2,V_Ga_2,T_b] :
      ( c_Hoare__Mirabelle_Ohoare__derivs(T_b,V_Ga_2,hAPP(hAPP(c_Set_Oinsert(tc_Hoare__Mirabelle_Otriple(T_b)),V_ta_2),c_Orderings_Obot__class_Obot(tc_fun(tc_Hoare__Mirabelle_Otriple(T_b),tc_HOL_Obool))))
     => ( c_Hoare__Mirabelle_Ohoare__derivs(T_b,V_Ga_2,V_tsa_2)
       => c_Hoare__Mirabelle_Ohoare__derivs(T_b,V_Ga_2,hAPP(hAPP(c_Set_Oinsert(tc_Hoare__Mirabelle_Otriple(T_b)),V_ta_2),V_tsa_2)) ) ) )).

fof(fact_empty__not__insert,axiom,(
    ! [V_A_2,V_a_2,T_b] : c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2) )).

fof(fact_insert__not__empty,axiom,(
    ! [V_A_2,V_a_2,T_b] : hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2) != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) )).

fof(fact_doubleton__eq__iff,axiom,(
    ! [V_d_2,V_c_2,V_b_2,V_a_2,T_b] :
      ( hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_b_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) = hAPP(hAPP(c_Set_Oinsert(T_b),V_c_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_d_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))
    <=> ( ( V_a_2 = V_c_2
          & V_b_2 = V_d_2 )
        | ( V_a_2 = V_d_2
          & V_b_2 = V_c_2 ) ) ) )).

fof(fact_singleton__inject,axiom,(
    ! [V_b_2,V_a_2,T_b] :
      ( hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) = hAPP(hAPP(c_Set_Oinsert(T_b),V_b_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))
     => V_a_2 = V_b_2 ) )).

fof(fact_bot__apply,axiom,(
    ! [V_x_2,T_c,T_b] :
      ( class_Orderings_Obot(T_b)
     => hAPP(c_Orderings_Obot__class_Obot(tc_fun(T_c,T_b)),V_x_2) = c_Orderings_Obot__class_Obot(T_b) ) )).

fof(fact_the__elem__eq,axiom,(
    ! [V_x_2,T_b] : c_Set_Othe__elem(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) = V_x_2 )).

fof(fact_insert__absorb2,axiom,(
    ! [V_A_2,V_x_2,T_b] : hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2) )).

fof(fact_insert__commute,axiom,(
    ! [V_A_2,V_y_2,V_x_2,T_b] : hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_y_2),V_A_2)) = hAPP(hAPP(c_Set_Oinsert(T_b),V_y_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) )).

fof(fact_insert__code,axiom,(
    ! [V_x_2,V_A_2,V_y_2,T_b] :
      ( hBOOL(hAPP(hAPP(hAPP(c_Set_Oinsert(T_b),V_y_2),V_A_2),V_x_2))
    <=> ( V_y_2 = V_x_2
        | hBOOL(hAPP(V_A_2,V_x_2)) ) ) )).

fof(fact_fold1Set__sing,axiom,(
    ! [V_b_2,V_a_2,V_f_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofold1Set(T_b,V_f_2,hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))),V_b_2))
    <=> V_a_2 = V_b_2 ) )).

fof(fact_folding__one_Osingleton,axiom,(
    ! [V_x_2,V_F_2,V_f_2,T_b] :
      ( c_Finite__Set_Ofolding__one(T_b,V_f_2,V_F_2)
     => hAPP(V_F_2,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) = V_x_2 ) )).

fof(fact_hoare__derivs_OSkip,axiom,(
    ! [V_P_2,V_Ga_2,T_b] : c_Hoare__Mirabelle_Ohoare__derivs(T_b,V_Ga_2,hAPP(hAPP(c_Set_Oinsert(tc_Hoare__Mirabelle_Otriple(T_b)),c_Hoare__Mirabelle_Otriple_Otriple(T_b,V_P_2,c_Com_Ocom_OSKIP,V_P_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_Hoare__Mirabelle_Otriple(T_b),tc_HOL_Obool)))) )).

fof(fact_triple_Oinject,axiom,(
    ! [V_fun2_H_2,V_com_H_2,V_fun1_H_2,V_fun2_2,V_com_2,V_fun1_2,T_b] :
      ( c_Hoare__Mirabelle_Otriple_Otriple(T_b,V_fun1_2,V_com_2,V_fun2_2) = c_Hoare__Mirabelle_Otriple_Otriple(T_b,V_fun1_H_2,V_com_H_2,V_fun2_H_2)
    <=> ( V_fun1_2 = V_fun1_H_2
        & V_com_2 = V_com_H_2
        & V_fun2_2 = V_fun2_H_2 ) ) )).

fof(fact_fold1Set__nonempty,axiom,(
    ! [V_x_2,V_A_2,V_f_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofold1Set(T_b,V_f_2,V_A_2),V_x_2))
     => V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) ) )).

fof(fact_empty__fold1SetE,axiom,(
    ! [V_x_2,V_f_2,T_b] : ~ hBOOL(hAPP(c_Finite__Set_Ofold1Set(T_b,V_f_2,c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))),V_x_2)) )).

fof(fact_hoare__derivs_Oequations_I7_J,axiom,(
    ! [V_a2_2,V_a1_2,T_b] : c_Hoare__Mirabelle_Ohoare__derivs(T_b,V_a1_2,hAPP(hAPP(c_Set_Oinsert(tc_Hoare__Mirabelle_Otriple(T_b)),c_Hoare__Mirabelle_Otriple_Otriple(T_b,V_a2_2,c_Com_Ocom_OSKIP,V_a2_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_Hoare__Mirabelle_Otriple(T_b),tc_HOL_Obool)))) )).

fof(fact_triple_Orecs,axiom,(
    ! [V_fun2_2,V_com_2,V_fun1_2,V_f1_2,T_b,T_c] : c_Hoare__Mirabelle_Otriple_Otriple__rec(T_c,T_b,V_f1_2,c_Hoare__Mirabelle_Otriple_Otriple(T_c,V_fun1_2,V_com_2,V_fun2_2)) = hAPP(hAPP(hAPP(V_f1_2,V_fun1_2),V_com_2),V_fun2_2) )).

fof(fact_triple_Osimps_I2_J,axiom,(
    ! [V_fun2_2,V_com_2,V_fun1_2,V_f1_2,T_b,T_c] : c_Hoare__Mirabelle_Otriple_Otriple__case(T_c,T_b,V_f1_2,c_Hoare__Mirabelle_Otriple_Otriple(T_c,V_fun1_2,V_com_2,V_fun2_2)) = hAPP(hAPP(hAPP(V_f1_2,V_fun1_2),V_com_2),V_fun2_2) )).

fof(fact_conseq1,axiom,(
    ! [V_P_2,V_Q_2,V_c_2,V_P_H_2,V_Ga_2,T_b] :
      ( c_Hoare__Mirabelle_Ohoare__derivs(T_b,V_Ga_2,hAPP(hAPP(c_Set_Oinsert(tc_Hoare__Mirabelle_Otriple(T_b)),c_Hoare__Mirabelle_Otriple_Otriple(T_b,V_P_H_2,V_c_2,V_Q_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_Hoare__Mirabelle_Otriple(T_b),tc_HOL_Obool))))
     => ( ! [B_Z,B_s] :
            ( hBOOL(hAPP(hAPP(V_P_2,B_Z),B_s))
           => hBOOL(hAPP(hAPP(V_P_H_2,B_Z),B_s)) )
       => c_Hoare__Mirabelle_Ohoare__derivs(T_b,V_Ga_2,hAPP(hAPP(c_Set_Oinsert(tc_Hoare__Mirabelle_Otriple(T_b)),c_Hoare__Mirabelle_Otriple_Otriple(T_b,V_P_2,V_c_2,V_Q_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_Hoare__Mirabelle_Otriple(T_b),tc_HOL_Obool)))) ) ) )).

fof(fact_conseq2,axiom,(
    ! [V_Q_2,V_Q_H_2,V_c_2,V_P_2,V_Ga_2,T_b] :
      ( c_Hoare__Mirabelle_Ohoare__derivs(T_b,V_Ga_2,hAPP(hAPP(c_Set_Oinsert(tc_Hoare__Mirabelle_Otriple(T_b)),c_Hoare__Mirabelle_Otriple_Otriple(T_b,V_P_2,V_c_2,V_Q_H_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_Hoare__Mirabelle_Otriple(T_b),tc_HOL_Obool))))
     => ( ! [B_Z,B_s] :
            ( hBOOL(hAPP(hAPP(V_Q_H_2,B_Z),B_s))
           => hBOOL(hAPP(hAPP(V_Q_2,B_Z),B_s)) )
       => c_Hoare__Mirabelle_Ohoare__derivs(T_b,V_Ga_2,hAPP(hAPP(c_Set_Oinsert(tc_Hoare__Mirabelle_Otriple(T_b)),c_Hoare__Mirabelle_Otriple_Otriple(T_b,V_P_2,V_c_2,V_Q_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_Hoare__Mirabelle_Otriple(T_b),tc_HOL_Obool)))) ) ) )).

fof(fact_Comp,axiom,(
    ! [V_R_2,V_d_2,V_Q_2,V_c_2,V_P_2,V_Ga_2,T_b] :
      ( c_Hoare__Mirabelle_Ohoare__derivs(T_b,V_Ga_2,hAPP(hAPP(c_Set_Oinsert(tc_Hoare__Mirabelle_Otriple(T_b)),c_Hoare__Mirabelle_Otriple_Otriple(T_b,V_P_2,V_c_2,V_Q_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_Hoare__Mirabelle_Otriple(T_b),tc_HOL_Obool))))
     => ( c_Hoare__Mirabelle_Ohoare__derivs(T_b,V_Ga_2,hAPP(hAPP(c_Set_Oinsert(tc_Hoare__Mirabelle_Otriple(T_b)),c_Hoare__Mirabelle_Otriple_Otriple(T_b,V_Q_2,V_d_2,V_R_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_Hoare__Mirabelle_Otriple(T_b),tc_HOL_Obool))))
       => c_Hoare__Mirabelle_Ohoare__derivs(T_b,V_Ga_2,hAPP(hAPP(c_Set_Oinsert(tc_Hoare__Mirabelle_Otriple(T_b)),c_Hoare__Mirabelle_Otriple_Otriple(T_b,V_P_2,c_Com_Ocom_OSemi(V_c_2,V_d_2),V_R_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_Hoare__Mirabelle_Otriple(T_b),tc_HOL_Obool)))) ) ) )).

fof(fact_conseq12,axiom,(
    ! [V_Q_2,V_P_2,V_Q_H_2,V_c_2,V_P_H_2,V_Ga_2,T_b] :
      ( c_Hoare__Mirabelle_Ohoare__derivs(T_b,V_Ga_2,hAPP(hAPP(c_Set_Oinsert(tc_Hoare__Mirabelle_Otriple(T_b)),c_Hoare__Mirabelle_Otriple_Otriple(T_b,V_P_H_2,V_c_2,V_Q_H_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_Hoare__Mirabelle_Otriple(T_b),tc_HOL_Obool))))
     => ( ! [B_Z,B_s] :
            ( hBOOL(hAPP(hAPP(V_P_2,B_Z),B_s))
           => ! [B_s_H] :
                ( ! [B_Z_H] :
                    ( hBOOL(hAPP(hAPP(V_P_H_2,B_Z_H),B_s))
                   => hBOOL(hAPP(hAPP(V_Q_H_2,B_Z_H),B_s_H)) )
               => hBOOL(hAPP(hAPP(V_Q_2,B_Z),B_s_H)) ) )
       => c_Hoare__Mirabelle_Ohoare__derivs(T_b,V_Ga_2,hAPP(hAPP(c_Set_Oinsert(tc_Hoare__Mirabelle_Otriple(T_b)),c_Hoare__Mirabelle_Otriple_Otriple(T_b,V_P_2,V_c_2,V_Q_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_Hoare__Mirabelle_Otriple(T_b),tc_HOL_Obool)))) ) ) )).

fof(fact_fold1__singleton__def,axiom,(
    ! [V_a_2,V_f_2,T_b,V_g_2] :
      ( V_g_2 = c_Finite__Set_Ofold1(T_b,V_f_2)
     => hAPP(V_g_2,hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) = V_a_2 ) )).

fof(fact_fold1__singleton,axiom,(
    ! [V_a_2,V_f_2,T_b] : hAPP(c_Finite__Set_Ofold1(T_b,V_f_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) = V_a_2 )).

fof(fact_subset__singletonD,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))
     => ( V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
        | V_A_2 = hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) ) ) )).

fof(fact_singletonE,axiom,(
    ! [V_a_2,V_b_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))
     => V_b_2 = V_a_2 ) )).

fof(fact_singleton__iff,axiom,(
    ! [V_a_2,V_b_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))
    <=> V_b_2 = V_a_2 ) )).

fof(fact_folding__image__simple_Oempty,axiom,(
    ! [V_F_2,V_g_2,V_z_2,V_f_2,T_c,T_b] :
      ( c_Finite__Set_Ofolding__image__simple(T_b,T_c,V_f_2,V_z_2,V_g_2,V_F_2)
     => hAPP(V_F_2,c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) = V_z_2 ) )).

fof(fact_order__refl,axiom,(
    ! [V_x,T_a] :
      ( class_Orderings_Opreorder(T_a)
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_x)) ) )).

fof(fact_equalityCE,axiom,(
    ! [V_c_2,T_b,V_B_2,V_A_2] :
      ( V_A_2 = V_B_2
     => ( ( hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_A_2))
         => ~ hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_B_2)) )
       => ~ ( ~ hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_A_2))
           => hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_B_2)) ) ) ) )).

fof(fact_equalityI,axiom,(
    ! [V_B_2,V_A_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2))
       => V_A_2 = V_B_2 ) ) )).

fof(fact_emptyE,axiom,(
    ! [V_a_2,T_b] : ~ hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) )).

fof(fact_insertE,axiom,(
    ! [V_A_2,V_b_2,V_a_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_b_2),V_A_2)))
     => ( V_a_2 != V_b_2
       => hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) ) ) )).

fof(fact_insertCI,axiom,(
    ! [V_b_2,V_B_2,V_a_2,T_b] :
      ( ( ~ hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_B_2))
       => V_a_2 = V_b_2 )
     => hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_b_2),V_B_2))) ) )).

fof(fact_subsetD,axiom,(
    ! [V_c_2,V_B_2,V_A_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))
     => ( hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_A_2))
       => hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_B_2)) ) ) )).

fof(fact_empty__subsetI,axiom,(
    ! [V_A_2,T_b] : hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))),V_A_2)) )).

fof(fact_subset__refl,axiom,(
    ! [V_A_2,T_b] : hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_A_2)) )).

fof(fact_le__fun__def,axiom,(
    ! [V_g_2,V_f_2,T_b,T_c] :
      ( class_Orderings_Oord(T_c)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,T_c)),V_f_2),V_g_2))
      <=> ! [B_x] : hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(V_f_2,B_x)),hAPP(V_g_2,B_x))) ) ) )).

fof(fact_mem__def,axiom,(
    ! [V_A_2,V_x_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
    <=> hBOOL(hAPP(V_A_2,V_x_2)) ) )).

fof(fact_linorder__linear,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y))
        | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),V_x)) ) ) )).

fof(fact_set__eq__subset,axiom,(
    ! [T_b,V_B_2,V_A_2] :
      ( V_A_2 = V_B_2
    <=> ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))
        & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2)) ) ) )).

fof(fact_order__eq__iff,axiom,(
    ! [V_y_2,V_x_2,T_b] :
      ( class_Orderings_Oorder(T_b)
     => ( V_x_2 = V_y_2
      <=> ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),V_y_2))
          & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_y_2),V_x_2)) ) ) ) )).

fof(fact_equalityD1,axiom,(
    ! [T_b,V_B_2,V_A_2] :
      ( V_A_2 = V_B_2
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) ) )).

fof(fact_equalityD2,axiom,(
    ! [T_b,V_B_2,V_A_2] :
      ( V_A_2 = V_B_2
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2)) ) )).

fof(fact_order__eq__refl,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Orderings_Opreorder(T_a)
     => ( V_x = V_y
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y)) ) ) )).

fof(fact_le__funD,axiom,(
    ! [V_x_2,V_g_2,V_f_2,T_b,T_c] :
      ( class_Orderings_Oord(T_c)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,T_c)),V_f_2),V_g_2))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(V_f_2,V_x_2)),hAPP(V_g_2,V_x_2))) ) ) )).

fof(fact_eqset__imp__iff,axiom,(
    ! [V_x_2,T_b,V_B_2,V_A_2] :
      ( V_A_2 = V_B_2
     => ( hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
      <=> hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_B_2)) ) ) )).

fof(fact_eqelem__imp__iff,axiom,(
    ! [V_A_2,T_b,V_y_2,V_x_2] :
      ( V_x_2 = V_y_2
     => ( hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
      <=> hBOOL(hAPP(hAPP(c_member(T_b),V_y_2),V_A_2)) ) ) )).

fof(fact_in__mono,axiom,(
    ! [V_x_2,V_B_2,V_A_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))
     => ( hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
       => hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_B_2)) ) ) )).

fof(fact_order__antisym__conv,axiom,(
    ! [V_x_2,V_y_2,T_b] :
      ( class_Orderings_Oorder(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_y_2),V_x_2))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),V_y_2))
        <=> V_x_2 = V_y_2 ) ) ) )).

fof(fact_eq__mem__trans,axiom,(
    ! [V_A_2,T_b,V_b_2,V_a_2] :
      ( V_a_2 = V_b_2
     => ( hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),V_A_2))
       => hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) ) ) )).

fof(fact_ord__eq__le__trans,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Orderings_Oord(T_a)
     => ( V_a = V_b
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),V_c))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_c)) ) ) ) )).

fof(fact_xt1_I3_J,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Orderings_Oorder(T_a)
     => ( V_a = V_b
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_c),V_b))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_c),V_a)) ) ) ) )).

fof(fact_set__rev__mp,axiom,(
    ! [V_B_2,V_A_2,V_x_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))
       => hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_B_2)) ) ) )).

fof(fact_set__mp,axiom,(
    ! [V_x_2,V_B_2,V_A_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))
     => ( hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
       => hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_B_2)) ) ) )).

fof(fact_subset__trans,axiom,(
    ! [V_C_2,V_B_2,V_A_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_C_2)) ) ) )).

fof(fact_ord__le__eq__trans,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Orderings_Oord(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b))
       => ( V_b = V_c
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_c)) ) ) ) )).

fof(fact_xt1_I4_J,axiom,(
    ! [V_c,V_a,V_b,T_a] :
      ( class_Orderings_Oorder(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),V_a))
       => ( V_b = V_c
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_c),V_a)) ) ) ) )).

fof(fact_order__antisym,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Orderings_Oorder(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),V_x))
         => V_x = V_y ) ) ) )).

fof(fact_order__trans,axiom,(
    ! [V_z,V_y,V_x,T_a] :
      ( class_Orderings_Opreorder(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),V_z))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_z)) ) ) ) )).

fof(fact_xt1_I5_J,axiom,(
    ! [V_x,V_y,T_a] :
      ( class_Orderings_Oorder(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),V_x))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y))
         => V_x = V_y ) ) ) )).

fof(fact_xt1_I6_J,axiom,(
    ! [V_z,V_x,V_y,T_a] :
      ( class_Orderings_Oorder(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),V_x))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_z),V_y))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_z),V_x)) ) ) ) )).

fof(fact_equalityE,axiom,(
    ! [T_b,V_B_2,V_A_2] :
      ( V_A_2 = V_B_2
     => ~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))
         => ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2)) ) ) )).

fof(fact_le__funE,axiom,(
    ! [V_x_2,V_g_2,V_f_2,T_b,T_c] :
      ( class_Orderings_Oord(T_c)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,T_c)),V_f_2),V_g_2))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(V_f_2,V_x_2)),hAPP(V_g_2,V_x_2))) ) ) )).

fof(fact_linorder__le__cases,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),V_x)) ) ) )).

fof(fact_subset__insert,axiom,(
    ! [V_B_2,V_A_2,V_x_2,T_b] :
      ( ~ hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_B_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) ) ) )).

fof(fact_insert__subset,axiom,(
    ! [V_B_2,V_A_2,V_x_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)),V_B_2))
    <=> ( hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_B_2))
        & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) ) ) )).

fof(fact_bot__least,axiom,(
    ! [V_x,T_a] :
      ( class_Orderings_Obot(T_a)
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Orderings_Obot__class_Obot(T_a)),V_x)) ) )).

fof(fact_all__not__in__conv,axiom,(
    ! [V_A_2,T_b] :
      ( ! [B_x] : ~ hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2))
    <=> V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) ) )).

fof(fact_ex__in__conv,axiom,(
    ! [V_A_2,T_b] :
      ( ? [B_x] : hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2))
    <=> V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) ) )).

fof(fact_empty__iff,axiom,(
    ! [V_c_2,T_b] : ~ hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) )).

fof(fact_equals0D,axiom,(
    ! [V_a_2,T_b,V_A_2] :
      ( V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
     => ~ hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) ) )).

fof(fact_subset__empty,axiom,(
    ! [V_A_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))
    <=> V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) ) )).

fof(fact_insertI1,axiom,(
    ! [V_B_2,V_a_2,T_b] : hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_B_2))) )).

fof(fact_insert__iff,axiom,(
    ! [V_A_2,V_b_2,V_a_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_b_2),V_A_2)))
    <=> ( V_a_2 = V_b_2
        | hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) ) ) )).

fof(fact_insert__ident,axiom,(
    ! [V_B_2,V_A_2,V_x_2,T_b] :
      ( ~ hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
     => ( ~ hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_B_2))
       => ( hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2) = hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_B_2)
        <=> V_A_2 = V_B_2 ) ) ) )).

fof(fact_insertI2,axiom,(
    ! [V_b_2,V_B_2,V_a_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_B_2))
     => hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_b_2),V_B_2))) ) )).

fof(fact_insert__absorb,axiom,(
    ! [V_A_2,V_a_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2))
     => hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2) = V_A_2 ) )).

fof(fact_subset__insertI,axiom,(
    ! [V_a_2,V_B_2,T_b] : hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_B_2))) )).

fof(fact_subset__insertI2,axiom,(
    ! [V_b_2,V_B_2,V_A_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_b_2),V_B_2))) ) )).

fof(fact_insert__mono,axiom,(
    ! [V_a_2,V_D_2,V_C_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_D_2))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_C_2)),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_D_2))) ) )).

fof(fact_asm,axiom,(
    ! [V_Ga_2,V_tsa_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Hoare__Mirabelle_Otriple(T_b),tc_HOL_Obool)),V_tsa_2),V_Ga_2))
     => c_Hoare__Mirabelle_Ohoare__derivs(T_b,V_Ga_2,V_tsa_2) ) )).

fof(fact_weaken,axiom,(
    ! [V_tsa_2,V_ts_H_2,V_Ga_2,T_b] :
      ( c_Hoare__Mirabelle_Ohoare__derivs(T_b,V_Ga_2,V_ts_H_2)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Hoare__Mirabelle_Otriple(T_b),tc_HOL_Obool)),V_tsa_2),V_ts_H_2))
       => c_Hoare__Mirabelle_Ohoare__derivs(T_b,V_Ga_2,V_tsa_2) ) ) )).

fof(fact_thin,axiom,(
    ! [V_Ga_2,V_tsa_2,V_G_H_2,T_b] :
      ( c_Hoare__Mirabelle_Ohoare__derivs(T_b,V_G_H_2,V_tsa_2)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Hoare__Mirabelle_Otriple(T_b),tc_HOL_Obool)),V_G_H_2),V_Ga_2))
       => c_Hoare__Mirabelle_Ohoare__derivs(T_b,V_Ga_2,V_tsa_2) ) ) )).

fof(fact_com_Osimps_I13_J,axiom,(
    ! [V_com2_H,V_com1_H] : c_Com_Ocom_OSemi(V_com1_H,V_com2_H) != c_Com_Ocom_OSKIP )).

fof(fact_com_Osimps_I12_J,axiom,(
    ! [V_com2_H,V_com1_H] : c_Com_Ocom_OSKIP != c_Com_Ocom_OSemi(V_com1_H,V_com2_H) )).

fof(fact_fold1Set_Ointros,axiom,(
    ! [V_x_2,V_A_2,V_a_2,V_f_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_b,T_b,V_f_2,V_a_2,V_A_2),V_x_2))
     => ( ~ hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2))
       => hBOOL(hAPP(c_Finite__Set_Ofold1Set(T_b,V_f_2,hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2)),V_x_2)) ) ) )).

fof(fact_folding__one_Oinsert,axiom,(
    ! [V_x_2,V_A_2,V_F_2,V_f_2,T_b] :
      ( c_Finite__Set_Ofolding__one(T_b,V_f_2,V_F_2)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( ~ hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
         => ( V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
           => hAPP(V_F_2,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(V_f_2,V_x_2),hAPP(V_F_2,V_A_2)) ) ) ) ) )).

fof(fact_com_Osimps_I3_J,axiom,(
    ! [V_com2_H_2,V_com1_H_2,V_com2_2,V_com1_2] :
      ( c_Com_Ocom_OSemi(V_com1_2,V_com2_2) = c_Com_Ocom_OSemi(V_com1_H_2,V_com2_H_2)
    <=> ( V_com1_2 = V_com1_H_2
        & V_com2_2 = V_com2_H_2 ) ) )).

fof(fact_subset__insert__iff,axiom,(
    ! [V_B_2,V_x_2,V_A_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_B_2)))
    <=> ( ( hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))),V_B_2)) )
        & ( ~ hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) ) ) ) )).

fof(fact_diff__single__insert,axiom,(
    ! [V_B_2,V_x_2,V_A_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))),V_B_2))
     => ( hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_B_2))) ) ) )).

fof(fact_nonempty__iff,axiom,(
    ! [T_b,V_A_2] :
      ( V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
    <=> ? [B_x,B_B] :
          ( V_A_2 = hAPP(hAPP(c_Set_Oinsert(T_b),B_x),B_B)
          & ~ hBOOL(hAPP(hAPP(c_member(T_b),B_x),B_B)) ) ) )).

fof(fact_folding__image__simple_Oinsert,axiom,(
    ! [V_x_2,V_A_2,V_F_2,V_g_2,V_z_2,V_f_2,T_c,T_b] :
      ( c_Finite__Set_Ofolding__image__simple(T_b,T_c,V_f_2,V_z_2,V_g_2,V_F_2)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2))
       => ( ~ hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2))
         => hAPP(V_F_2,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)) = hAPP(hAPP(V_f_2,hAPP(V_g_2,V_x_2)),hAPP(V_F_2,V_A_2)) ) ) ) )).

fof(fact_predicate1D,axiom,(
    ! [V_x_2,V_Q_2,V_P_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_P_2),V_Q_2))
     => ( hBOOL(hAPP(V_P_2,V_x_2))
       => hBOOL(hAPP(V_Q_2,V_x_2)) ) ) )).

fof(fact_rev__predicate1D,axiom,(
    ! [V_Q_2,T_b,V_x_2,V_P_2] :
      ( hBOOL(hAPP(V_P_2,V_x_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_P_2),V_Q_2))
       => hBOOL(hAPP(V_Q_2,V_x_2)) ) ) )).

fof(fact_subsetI,axiom,(
    ! [V_B_2,V_A_2,T_b] :
      ( ! [B_x] :
          ( hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2))
         => hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_B_2)) )
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) ) )).

fof(fact_finite_OemptyI,axiom,(
    ! [T_b] : hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) )).

fof(fact_finite_OinsertI,axiom,(
    ! [V_a_2,V_A_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
     => hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2))) ) )).

fof(fact_DiffE,axiom,(
    ! [V_B_2,V_A_2,V_c_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)))
     => ~ ( hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_A_2))
         => hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_B_2)) ) ) )).

fof(fact_DiffI,axiom,(
    ! [V_B_2,V_A_2,V_c_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_A_2))
     => ( ~ hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_B_2))
       => hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) ) ) )).

fof(fact_finite__Diff,axiom,(
    ! [V_B_2,V_A_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
     => hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) ) )).

fof(fact_fold__graph__imp__finite,axiom,(
    ! [V_x_2,V_A_2,V_z_2,V_f_2,T_c,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_b,T_c,V_f_2,V_z_2,V_A_2),V_x_2))
     => hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) ) )).

fof(fact_finite__Diff2,axiom,(
    ! [V_A_2,V_B_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2))
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)))
      <=> hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) ) ) )).

fof(fact_Diff__idemp,axiom,(
    ! [V_B_2,V_A_2,T_b] : hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),V_B_2) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) )).

fof(fact_finite__code,axiom,(
    ! [V_A_2,T_b] :
      ( class_Finite__Set_Ofinite(T_b)
     => hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) ) )).

fof(fact_finite,axiom,(
    ! [V_A_2,T_b] :
      ( class_Finite__Set_Ofinite(T_b)
     => hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) ) )).

fof(fact_finite__Diff__insert,axiom,(
    ! [V_B_2,V_a_2,V_A_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_B_2))))
    <=> hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) ) )).

fof(fact_DiffD2,axiom,(
    ! [V_B_2,V_A_2,V_c_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)))
     => ~ hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_B_2)) ) )).

fof(fact_DiffD1,axiom,(
    ! [V_B_2,V_A_2,V_c_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)))
     => hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_A_2)) ) )).

fof(fact_Diff__iff,axiom,(
    ! [V_B_2,V_A_2,V_c_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)))
    <=> ( hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_A_2))
        & ~ hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_B_2)) ) ) )).

fof(fact_Diff__cancel,axiom,(
    ! [V_A_2,T_b] : hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_A_2) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) )).

fof(fact_Diff__empty,axiom,(
    ! [V_A_2,T_b] : hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) = V_A_2 )).

fof(fact_empty__Diff,axiom,(
    ! [V_A_2,T_b] : hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))),V_A_2) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) )).

fof(fact_double__diff,axiom,(
    ! [V_C_2,V_B_2,V_A_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2))
       => hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_B_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_A_2)) = V_A_2 ) ) )).

fof(fact_Diff__mono,axiom,(
    ! [V_B_2,V_D_2,V_C_2,V_A_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_C_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_D_2),V_B_2))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_D_2))) ) ) )).

fof(fact_Diff__subset,axiom,(
    ! [V_B_2,V_A_2,T_b] : hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),V_A_2)) )).

fof(fact_finite_Oequations_I1_J,axiom,(
    ! [T_b] : hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) )).

fof(fact_finite__insert,axiom,(
    ! [V_A_2,V_a_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2)))
    <=> hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) ) )).

fof(fact_finite__subset,axiom,(
    ! [V_B_2,V_A_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2))
       => hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) ) ) )).

fof(fact_rev__finite__subset,axiom,(
    ! [V_A_2,V_B_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))
       => hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) ) ) )).

fof(fact_empty__fold__graphE,axiom,(
    ! [V_x_2,V_z_2,V_f_2,T_c,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_b,T_c,V_f_2,V_z_2,c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))),V_x_2))
     => V_x_2 = V_z_2 ) )).

fof(fact_fold__graph_OemptyI,axiom,(
    ! [V_z_2,V_f_2,T_c,T_b] : hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_b,T_c,V_f_2,V_z_2,c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))),V_z_2)) )).

fof(fact_fold__graph_Oequations_I1_J,axiom,(
    ! [V_z_2,V_f_2,T_c,T_b] : hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_b,T_c,V_f_2,V_z_2,c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))),V_z_2)) )).

fof(fact_Diff1__fold__graph,axiom,(
    ! [V_y_2,V_x_2,V_A_2,V_z_2,V_f_2,T_c,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_b,T_c,V_f_2,V_z_2,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))),V_y_2))
     => ( hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
       => hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_b,T_c,V_f_2,V_z_2,V_A_2),hAPP(hAPP(V_f_2,V_x_2),V_y_2))) ) ) )).

fof(fact_folding__one_Oinsert__remove,axiom,(
    ! [V_x_2,V_A_2,V_F_2,V_f_2,T_b] :
      ( c_Finite__Set_Ofolding__one(T_b,V_f_2,V_F_2)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( ( hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
           => hAPP(V_F_2,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = V_x_2 )
          & ( hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
           => hAPP(V_F_2,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(V_f_2,V_x_2),hAPP(V_F_2,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))) ) ) ) ) )).

fof(fact_folding__image__simple_Oinsert__remove,axiom,(
    ! [V_x_2,V_A_2,V_F_2,V_g_2,V_z_2,V_f_2,T_c,T_b] :
      ( c_Finite__Set_Ofolding__image__simple(T_b,T_c,V_f_2,V_z_2,V_g_2,V_F_2)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2))
       => hAPP(V_F_2,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)) = hAPP(hAPP(V_f_2,hAPP(V_g_2,V_x_2)),hAPP(V_F_2,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))))) ) ) )).

fof(fact_insert__Diff__if,axiom,(
    ! [V_A_2,V_B_2,V_x_2,T_b] :
      ( ( hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_B_2))
       => hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)),V_B_2) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) )
      & ( ~ hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_B_2))
       => hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)),V_B_2) = hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) ) ) )).

fof(fact_insert__Diff1,axiom,(
    ! [V_A_2,V_B_2,V_x_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_B_2))
     => hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)),V_B_2) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) ) )).

fof(fact_insert__Diff__single,axiom,(
    ! [V_A_2,V_a_2,T_b] : hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))) = hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2) )).

fof(fact_Diff__insert2,axiom,(
    ! [V_B_2,V_a_2,V_A_2,T_b] : hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_B_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))),V_B_2) )).

fof(fact_Diff__insert,axiom,(
    ! [V_B_2,V_a_2,V_A_2,T_b] : hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_B_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) )).

fof(fact_folding__one_Oremove,axiom,(
    ! [V_x_2,V_A_2,V_F_2,V_f_2,T_b] :
      ( c_Finite__Set_Ofolding__one(T_b,V_f_2,V_F_2)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
         => ( ( hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
             => hAPP(V_F_2,V_A_2) = V_x_2 )
            & ( hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
             => hAPP(V_F_2,V_A_2) = hAPP(hAPP(V_f_2,V_x_2),hAPP(V_F_2,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))) ) ) ) ) ) )).

fof(fact_folding__image__simple_Oremove,axiom,(
    ! [V_x_2,V_A_2,V_F_2,V_g_2,V_z_2,V_f_2,T_c,T_b] :
      ( c_Finite__Set_Ofolding__image__simple(T_b,T_c,V_f_2,V_z_2,V_g_2,V_F_2)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2))
       => ( hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2))
         => hAPP(V_F_2,V_A_2) = hAPP(hAPP(V_f_2,hAPP(V_g_2,V_x_2)),hAPP(V_F_2,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))))) ) ) ) )).

fof(fact_fold__graph_OinsertI,axiom,(
    ! [V_y_2,V_z_2,V_f_2,T_c,V_A_2,V_x_2,T_b] :
      ( ~ hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
     => ( hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_b,T_c,V_f_2,V_z_2,V_A_2),V_y_2))
       => hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_b,T_c,V_f_2,V_z_2,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)),hAPP(hAPP(V_f_2,V_x_2),V_y_2))) ) ) )).

fof(fact_folding__one_Oeq__fold,axiom,(
    ! [V_A_2,V_F_2,V_f_2,T_b] :
      ( c_Finite__Set_Ofolding__one(T_b,V_f_2,V_F_2)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => hAPP(V_F_2,V_A_2) = hAPP(c_Finite__Set_Ofold1(T_b,V_f_2),V_A_2) ) ) )).

fof(fact_insert__Diff,axiom,(
    ! [V_A_2,V_a_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2))
     => hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))) = V_A_2 ) )).

fof(fact_Diff__insert__absorb,axiom,(
    ! [V_A_2,V_x_2,T_b] :
      ( ~ hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
     => hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) = V_A_2 ) )).

fof(fact_eq__mem,axiom,(
    ! [V_y_2,V_x_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_fequal,V_y_2)))
    <=> V_x_2 = V_y_2 ) )).

fof(fact_minus__apply,axiom,(
    ! [V_x_2,V_B_2,V_A_2,T_c,T_b] :
      ( class_Groups_Ominus(T_b)
     => hAPP(hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,T_b)),V_A_2),V_B_2),V_x_2) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),hAPP(V_A_2,V_x_2)),hAPP(V_B_2,V_x_2)) ) )).

fof(fact_diff__eq__diff__less__eq,axiom,(
    ! [V_d_2,V_c_2,V_b_2,V_a_2,T_b] :
      ( class_Groups_Oordered__ab__group__add(T_b)
     => ( hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),V_a_2),V_b_2) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),V_c_2),V_d_2)
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),V_b_2))
        <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_c_2),V_d_2)) ) ) ) )).

fof(fact_folding__one_Oclosed,axiom,(
    ! [V_A_2,V_F_2,V_f_2,T_b] :
      ( c_Finite__Set_Ofolding__one(T_b,V_f_2,V_F_2)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
         => ( ! [B_x,B_y] : hBOOL(hAPP(hAPP(c_member(T_b),hAPP(hAPP(V_f_2,B_x),B_y)),hAPP(hAPP(c_Set_Oinsert(T_b),B_x),hAPP(hAPP(c_Set_Oinsert(T_b),B_y),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))))
           => hBOOL(hAPP(hAPP(c_member(T_b),hAPP(V_F_2,V_A_2)),V_A_2)) ) ) ) ) )).

fof(fact_finite__empty__induct,axiom,(
    ! [V_P_2,V_A_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
     => ( hBOOL(hAPP(V_P_2,V_A_2))
       => ( ! [B_a,B_A] :
              ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),B_A))
             => ( hBOOL(hAPP(hAPP(c_member(T_b),B_a),B_A))
               => ( hBOOL(hAPP(V_P_2,B_A))
                 => hBOOL(hAPP(V_P_2,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),B_A),hAPP(hAPP(c_Set_Oinsert(T_b),B_a),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))) ) ) )
         => hBOOL(hAPP(V_P_2,c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) ) ) ) )).

fof(fact_finite__subset__induct,axiom,(
    ! [V_P_2,V_A_2,V_F_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_F_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_F_2),V_A_2))
       => ( hBOOL(hAPP(V_P_2,c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))
         => ( ! [B_a,B_F] :
                ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),B_F))
               => ( hBOOL(hAPP(hAPP(c_member(T_b),B_a),V_A_2))
                 => ( ~ hBOOL(hAPP(hAPP(c_member(T_b),B_a),B_F))
                   => ( hBOOL(hAPP(V_P_2,B_F))
                     => hBOOL(hAPP(V_P_2,hAPP(hAPP(c_Set_Oinsert(T_b),B_a),B_F))) ) ) ) )
           => hBOOL(hAPP(V_P_2,V_F_2)) ) ) ) ) )).

fof(fact_folding__one__idem_Osubset__idem,axiom,(
    ! [V_B_2,V_A_2,V_F_2,V_f_2,T_b] :
      ( c_Finite__Set_Ofolding__one__idem(T_b,V_f_2,V_F_2)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( V_B_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
         => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2))
           => hAPP(hAPP(V_f_2,hAPP(V_F_2,V_B_2)),hAPP(V_F_2,V_A_2)) = hAPP(V_F_2,V_A_2) ) ) ) ) )).

fof(fact_folding__one__idem_Oinsert__idem,axiom,(
    ! [V_x_2,V_A_2,V_F_2,V_f_2,T_b] :
      ( c_Finite__Set_Ofolding__one__idem(T_b,V_f_2,V_F_2)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
         => hAPP(V_F_2,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(V_f_2,V_x_2),hAPP(V_F_2,V_A_2)) ) ) ) )).

fof(fact_fold1Set_Oequations,axiom,(
    ! [V_a3_2,V_a2_2,V_a1_2,V_f_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofold1Set(T_b,V_f_2,hAPP(hAPP(c_Set_Oinsert(T_b),V_a1_2),V_a2_2)),V_a3_2))
    <=> ? [B_a,B_A] :
          ( hAPP(hAPP(c_Set_Oinsert(T_b),V_a1_2),V_a2_2) = hAPP(hAPP(c_Set_Oinsert(T_b),B_a),B_A)
          & hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_b,T_b,V_f_2,B_a,B_A),V_a3_2))
          & ~ hBOOL(hAPP(hAPP(c_member(T_b),B_a),B_A)) ) ) )).

fof(fact_insert__fold1SetE,axiom,(
    ! [V_x_2,V_X_2,V_a_2,V_f_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofold1Set(T_b,V_f_2,hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_X_2)),V_x_2))
     => ~ ! [B_a,B_A] :
            ( hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_X_2) = hAPP(hAPP(c_Set_Oinsert(T_b),B_a),B_A)
           => ( hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_b,T_b,V_f_2,B_a,B_A),V_x_2))
             => hBOOL(hAPP(hAPP(c_member(T_b),B_a),B_A)) ) ) ) )).

fof(fact_fold__graph__permute__diff,axiom,(
    ! [V_a_2,V_x_2,V_A_2,V_b_2,T_b] :
      ( class_Groups_Oab__semigroup__mult(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_b,T_b,c_Groups_Otimes__class_Otimes(T_b),V_b_2,V_A_2),V_x_2))
       => ( hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2))
         => ( ~ hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),V_A_2))
           => hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_b,T_b,c_Groups_Otimes__class_Otimes(T_b),V_a_2,hAPP(hAPP(c_Set_Oinsert(T_b),V_b_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))),V_x_2)) ) ) ) ) )).

fof(fact_fold1__insert,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( class_Groups_Oab__semigroup__mult(T_b)
     => ( V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
       => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
         => ( ~ hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
           => hAPP(c_Finite__Set_Ofold1(T_b,c_Groups_Otimes__class_Otimes(T_b)),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_x_2),hAPP(c_Finite__Set_Ofold1(T_b,c_Groups_Otimes__class_Otimes(T_b)),V_A_2)) ) ) ) ) )).

fof(fact_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Groups_Oab__semigroup__mult(T_a)
     => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)),V_c) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c)) ) )).

fof(fact_mult__left__idem,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Lattices_Oab__semigroup__idem__mult(T_a)
     => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b) ) )).

fof(fact_mult__idem,axiom,(
    ! [V_x,T_a] :
      ( class_Lattices_Oab__semigroup__idem__mult(T_a)
     => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_x) = V_x ) )).

fof(fact_times_Oidem,axiom,(
    ! [V_a,T_a] :
      ( class_Lattices_Oab__semigroup__idem__mult(T_a)
     => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_a) = V_a ) )).

fof(fact_folding__one__idem_Oidem,axiom,(
    ! [V_x_2,V_F_2,V_f_2,T_b] :
      ( c_Finite__Set_Ofolding__one__idem(T_b,V_f_2,V_F_2)
     => hAPP(hAPP(V_f_2,V_x_2),V_x_2) = V_x_2 ) )).

fof(fact_fold__graph__insert__swap,axiom,(
    ! [V_z_2,V_y_2,V_A_2,V_b_2,T_b] :
      ( class_Groups_Oab__semigroup__mult(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_b,T_b,c_Groups_Otimes__class_Otimes(T_b),V_b_2,V_A_2),V_y_2))
       => ( ~ hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),V_A_2))
         => hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_b,T_b,c_Groups_Otimes__class_Otimes(T_b),V_z_2,hAPP(hAPP(c_Set_Oinsert(T_b),V_b_2),V_A_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_z_2),V_y_2))) ) ) ) )).

fof(fact_diff__eq__diff__eq,axiom,(
    ! [V_d_2,V_c_2,V_b_2,V_a_2,T_b] :
      ( class_Groups_Oab__group__add(T_b)
     => ( hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),V_a_2),V_b_2) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),V_c_2),V_d_2)
       => ( V_a_2 = V_b_2
        <=> V_c_2 = V_d_2 ) ) ) )).

fof(fact_folding__one__idem_Oin__idem,axiom,(
    ! [V_x_2,V_A_2,V_F_2,V_f_2,T_b] :
      ( c_Finite__Set_Ofolding__one__idem(T_b,V_f_2,V_F_2)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
         => hAPP(hAPP(V_f_2,V_x_2),hAPP(V_F_2,V_A_2)) = hAPP(V_F_2,V_A_2) ) ) ) )).

fof(fact_fold1__insert__idem,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( class_Lattices_Oab__semigroup__idem__mult(T_b)
     => ( V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
       => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
         => hAPP(c_Finite__Set_Ofold1(T_b,c_Groups_Otimes__class_Otimes(T_b)),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_x_2),hAPP(c_Finite__Set_Ofold1(T_b,c_Groups_Otimes__class_Otimes(T_b)),V_A_2)) ) ) ) )).

fof(fact_fold1__in,axiom,(
    ! [V_A_2,T_b] :
      ( class_Groups_Oab__semigroup__mult(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
         => ( ! [B_x,B_y] : hBOOL(hAPP(hAPP(c_member(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),B_x),B_y)),hAPP(hAPP(c_Set_Oinsert(T_b),B_x),hAPP(hAPP(c_Set_Oinsert(T_b),B_y),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))))
           => hBOOL(hAPP(hAPP(c_member(T_b),hAPP(c_Finite__Set_Ofold1(T_b,c_Groups_Otimes__class_Otimes(T_b)),V_A_2)),V_A_2)) ) ) ) ) )).

fof(fact_psubset__insert__iff,axiom,(
    ! [V_B_2,V_x_2,V_A_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_B_2)))
    <=> ( ( hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_B_2))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) )
        & ( ~ hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_B_2))
         => ( ( hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
             => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))),V_B_2)) )
            & ( ~ hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
             => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) ) ) ) ) ) )).

fof(fact_folding__image__simple__idem_Osubset__idem,axiom,(
    ! [V_B_2,V_A_2,V_F_2,V_g_2,V_z_2,V_f_2,T_c,T_b] :
      ( c_Finite__Set_Ofolding__image__simple__idem(T_b,T_c,V_f_2,V_z_2,V_g_2,V_F_2)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_A_2))
         => hAPP(hAPP(V_f_2,hAPP(V_F_2,V_B_2)),hAPP(V_F_2,V_A_2)) = hAPP(V_F_2,V_A_2) ) ) ) )).

fof(fact_fold1__eq__fold,axiom,(
    ! [V_a_2,V_A_2,T_b] :
      ( class_Groups_Oab__semigroup__mult(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( ~ hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2))
         => hAPP(c_Finite__Set_Ofold1(T_b,c_Groups_Otimes__class_Otimes(T_b)),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2)) = hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Groups_Otimes__class_Otimes(T_b)),V_a_2),V_A_2) ) ) ) )).

fof(fact_folding__image__simple__idem_Oinsert__idem,axiom,(
    ! [V_x_2,V_A_2,V_F_2,V_g_2,V_z_2,V_f_2,T_c,T_b] :
      ( c_Finite__Set_Ofolding__image__simple__idem(T_b,T_c,V_f_2,V_z_2,V_g_2,V_F_2)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2))
       => hAPP(V_F_2,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)) = hAPP(hAPP(V_f_2,hAPP(V_g_2,V_x_2)),hAPP(V_F_2,V_A_2)) ) ) )).

fof(fact_finite__induct,axiom,(
    ! [V_P_2,V_F_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_F_2))
     => ( hBOOL(hAPP(V_P_2,c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))
       => ( ! [B_x,B_F] :
              ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),B_F))
             => ( ~ hBOOL(hAPP(hAPP(c_member(T_b),B_x),B_F))
               => ( hBOOL(hAPP(V_P_2,B_F))
                 => hBOOL(hAPP(V_P_2,hAPP(hAPP(c_Set_Oinsert(T_b),B_x),B_F))) ) ) )
         => hBOOL(hAPP(V_P_2,V_F_2)) ) ) ) )).

fof(fact_fold__graph_H_Ointros_I2_J,axiom,(
    ! [V_y_2,V_z_2,V_f_2,T_c,V_A_2,V_x_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
     => ( c_Nitpick_Ofold__graph_H(T_b,T_c,V_f_2,V_z_2,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))),V_y_2)
       => c_Nitpick_Ofold__graph_H(T_b,T_c,V_f_2,V_z_2,V_A_2,hAPP(hAPP(V_f_2,V_x_2),V_y_2)) ) ) )).

fof(fact_folding__image__simple__idem_Oin__idem,axiom,(
    ! [V_x_2,V_A_2,V_F_2,V_g_2,V_z_2,V_f_2,T_c,T_b] :
      ( c_Finite__Set_Ofolding__image__simple__idem(T_b,T_c,V_f_2,V_z_2,V_g_2,V_F_2)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2))
       => ( hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2))
         => hAPP(hAPP(V_f_2,hAPP(V_g_2,V_x_2)),hAPP(V_F_2,V_A_2)) = hAPP(V_F_2,V_A_2) ) ) ) )).

fof(fact_semilattice__big_OF__eq,axiom,(
    ! [V_A_2,V_F_2,V_f_2,T_b] :
      ( c_Big__Operators_Osemilattice__big(T_b,V_f_2,V_F_2)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => hAPP(V_F_2,V_A_2) = hAPP(c_Finite__Set_Ofold1(T_b,V_f_2),V_A_2) ) ) )).

fof(fact_fold__image__insert,axiom,(
    ! [V_z_2,V_g_2,V_a_2,V_A_2,T_b,T_c] :
      ( class_Groups_Oab__semigroup__mult(T_c)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( ~ hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2))
         => hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_c,T_b,c_Groups_Otimes__class_Otimes(T_c)),V_g_2),V_z_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(V_g_2,V_a_2)),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_c,T_b,c_Groups_Otimes__class_Otimes(T_c)),V_g_2),V_z_2),V_A_2)) ) ) ) )).

fof(fact_finite__nonempty__imp__fold1Set,axiom,(
    ! [V_f_2,V_A_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
     => ( V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
       => ? [B_x1] : hBOOL(hAPP(c_Finite__Set_Ofold1Set(T_b,V_f_2,V_A_2),B_x1)) ) ) )).

fof(fact_linorder__cases,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y))
       => ( V_x != V_y
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x)) ) ) ) )).

fof(fact_order__less__asym,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Orderings_Opreorder(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y))
       => ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x)) ) ) )).

fof(fact_xt1_I10_J,axiom,(
    ! [V_z,V_x,V_y,T_a] :
      ( class_Orderings_Oorder(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_z),V_y))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_z),V_x)) ) ) ) )).

fof(fact_order__less__trans,axiom,(
    ! [V_z,V_y,V_x,T_a] :
      ( class_Orderings_Opreorder(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_z))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_z)) ) ) ) )).

fof(fact_xt1_I2_J,axiom,(
    ! [V_c,V_a,V_b,T_a] :
      ( class_Orderings_Oorder(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),V_a))
       => ( V_b = V_c
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_c),V_a)) ) ) ) )).

fof(fact_ord__less__eq__trans,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Orderings_Oord(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b))
       => ( V_b = V_c
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_c)) ) ) ) )).

fof(fact_psubset__trans,axiom,(
    ! [V_C_2,V_B_2,V_A_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_C_2)) ) ) )).

fof(fact_xt1_I1_J,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Orderings_Oorder(T_a)
     => ( V_a = V_b
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_c),V_b))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_c),V_a)) ) ) ) )).

fof(fact_ord__eq__less__trans,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Orderings_Oord(T_a)
     => ( V_a = V_b
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),V_c))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_c)) ) ) ) )).

fof(fact_xt1_I9_J,axiom,(
    ! [V_a,V_b,T_a] :
      ( class_Orderings_Oorder(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),V_a))
       => ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)) ) ) )).

fof(fact_order__less__asym_H,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Orderings_Opreorder(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b))
       => ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),V_a)) ) ) )).

fof(fact_order__less__imp__not__eq2,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Orderings_Oorder(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y))
       => V_y != V_x ) ) )).

fof(fact_order__less__imp__not__eq,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Orderings_Oorder(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y))
       => V_x != V_y ) ) )).

fof(fact_order__less__imp__not__less,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Orderings_Opreorder(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y))
       => ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x)) ) ) )).

fof(fact_order__less__not__sym,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Orderings_Opreorder(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y))
       => ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x)) ) ) )).

fof(fact_less__imp__neq,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Orderings_Oorder(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y))
       => V_x != V_y ) ) )).

fof(fact_linorder__neqE,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => ( V_x != V_y
       => ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x)) ) ) ) )).

fof(fact_linorder__antisym__conv3,axiom,(
    ! [V_x_2,V_y_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_y_2),V_x_2))
       => ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_x_2),V_y_2))
        <=> V_x_2 = V_y_2 ) ) ) )).

fof(fact_linorder__less__linear,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y))
        | V_x = V_y
        | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x)) ) ) )).

fof(fact_not__less__iff__gr__or__eq,axiom,(
    ! [V_y_2,V_x_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_x_2),V_y_2))
      <=> ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_y_2),V_x_2))
          | V_x_2 = V_y_2 ) ) ) )).

fof(fact_linorder__neq__iff,axiom,(
    ! [V_y_2,V_x_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( V_x_2 != V_y_2
      <=> ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_x_2),V_y_2))
          | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_y_2),V_x_2)) ) ) ) )).

fof(fact_order__less__irrefl,axiom,(
    ! [V_x,T_a] :
      ( class_Orderings_Opreorder(T_a)
     => ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_x)) ) )).

fof(fact_folding__image__simple__idem_Oidem,axiom,(
    ! [V_x_2,V_F_2,V_g_2,V_z_2,V_f_2,T_c,T_b] :
      ( c_Finite__Set_Ofolding__image__simple__idem(T_b,T_c,V_f_2,V_z_2,V_g_2,V_F_2)
     => hAPP(hAPP(V_f_2,V_x_2),V_x_2) = V_x_2 ) )).

fof(fact_fold__graph_H_Ointros_I1_J,axiom,(
    ! [V_z_2,V_f_2,T_c,T_b] : c_Nitpick_Ofold__graph_H(T_b,T_c,V_f_2,V_z_2,c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)),V_z_2) )).

fof(fact_fold__graph_H_Oequations_I1_J,axiom,(
    ! [V_a2_2,V_a1_2,T_c,T_b] : c_Nitpick_Ofold__graph_H(T_b,T_c,V_a1_2,V_a2_2,c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)),V_a2_2) )).

fof(fact_fold__empty,axiom,(
    ! [V_z_2,V_f_2,T_b,T_c] : hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_b,V_f_2),V_z_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) = V_z_2 )).

fof(fact_less__fun__def,axiom,(
    ! [V_g_2,V_f_2,T_b,T_c] :
      ( class_Orderings_Oord(T_c)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_b,T_c)),V_f_2),V_g_2))
      <=> ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,T_c)),V_f_2),V_g_2))
          & ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,T_c)),V_g_2),V_f_2)) ) ) ) )).

fof(fact_xt1_I8_J,axiom,(
    ! [V_z,V_x,V_y,T_a] :
      ( class_Orderings_Oorder(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),V_x))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_z),V_y))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_z),V_x)) ) ) ) )).

fof(fact_order__le__less__trans,axiom,(
    ! [V_z,V_y,V_x,T_a] :
      ( class_Orderings_Opreorder(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_z))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_z)) ) ) ) )).

fof(fact_xt1_I7_J,axiom,(
    ! [V_z,V_x,V_y,T_a] :
      ( class_Orderings_Oorder(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_z),V_y))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_z),V_x)) ) ) ) )).

fof(fact_order__less__le__trans,axiom,(
    ! [V_z,V_y,V_x,T_a] :
      ( class_Orderings_Opreorder(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),V_z))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_z)) ) ) ) )).

fof(fact_xt1_I11_J,axiom,(
    ! [V_a,V_b,T_a] :
      ( class_Orderings_Oorder(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),V_a))
       => ( V_a != V_b
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),V_a)) ) ) ) )).

fof(fact_order__le__neq__trans,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Orderings_Oorder(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b))
       => ( V_a != V_b
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)) ) ) ) )).

fof(fact_order__le__imp__less__or__eq,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Orderings_Oorder(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y))
          | V_x = V_y ) ) ) )).

fof(fact_linorder__antisym__conv2,axiom,(
    ! [V_y_2,V_x_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),V_y_2))
       => ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_x_2),V_y_2))
        <=> V_x_2 = V_y_2 ) ) ) )).

fof(fact_order__less__imp__le,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Orderings_Opreorder(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y)) ) ) )).

fof(fact_leD,axiom,(
    ! [V_x,V_y,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),V_x))
       => ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y)) ) ) )).

fof(fact_xt1_I12_J,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Orderings_Oorder(T_a)
     => ( V_a != V_b
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),V_a))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),V_a)) ) ) ) )).

fof(fact_order__neq__le__trans,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Orderings_Oorder(T_a)
     => ( V_a != V_b
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)) ) ) ) )).

fof(fact_linorder__antisym__conv1,axiom,(
    ! [V_y_2,V_x_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_x_2),V_y_2))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),V_y_2))
        <=> V_x_2 = V_y_2 ) ) ) )).

fof(fact_not__leE,axiom,(
    ! [V_x,V_y,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),V_x))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y)) ) ) )).

fof(fact_leI,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),V_x)) ) ) )).

fof(fact_order__le__less,axiom,(
    ! [V_y_2,V_x_2,T_b] :
      ( class_Orderings_Oorder(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),V_y_2))
      <=> ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_x_2),V_y_2))
          | V_x_2 = V_y_2 ) ) ) )).

fof(fact_less__le__not__le,axiom,(
    ! [V_y_2,V_x_2,T_b] :
      ( class_Orderings_Opreorder(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_x_2),V_y_2))
      <=> ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),V_y_2))
          & ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_y_2),V_x_2)) ) ) ) )).

fof(fact_order__less__le,axiom,(
    ! [V_y_2,V_x_2,T_b] :
      ( class_Orderings_Oorder(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_x_2),V_y_2))
      <=> ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),V_y_2))
          & V_x_2 != V_y_2 ) ) ) )).

fof(fact_linorder__le__less__linear,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y))
        | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x)) ) ) )).

fof(fact_linorder__not__le,axiom,(
    ! [V_y_2,V_x_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),V_y_2))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_y_2),V_x_2)) ) ) )).

fof(fact_linorder__not__less,axiom,(
    ! [V_y_2,V_x_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_x_2),V_y_2))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_y_2),V_x_2)) ) ) )).

fof(fact_psubsetD,axiom,(
    ! [V_c_2,V_B_2,V_A_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))
     => ( hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_A_2))
       => hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_B_2)) ) ) )).

fof(fact_not__psubset__empty,axiom,(
    ! [V_A_2,T_b] : ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_b,tc_HOL_Obool)),V_A_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) )).

fof(fact_diff__eq__diff__less,axiom,(
    ! [V_d_2,V_c_2,V_b_2,V_a_2,T_b] :
      ( class_Groups_Oordered__ab__group__add(T_b)
     => ( hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),V_a_2),V_b_2) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),V_c_2),V_d_2)
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),V_b_2))
        <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),V_d_2)) ) ) ) )).

fof(fact_subset__psubset__trans,axiom,(
    ! [V_C_2,V_B_2,V_A_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_C_2)) ) ) )).

fof(fact_psubset__subset__trans,axiom,(
    ! [V_C_2,V_B_2,V_A_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_C_2)) ) ) )).

fof(fact_psubset__imp__subset,axiom,(
    ! [V_B_2,V_A_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) ) )).

fof(fact_subset__iff__psubset__eq,axiom,(
    ! [V_B_2,V_A_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))
    <=> ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))
        | V_A_2 = V_B_2 ) ) )).

fof(fact_psubset__eq,axiom,(
    ! [V_B_2,V_A_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))
    <=> ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))
        & V_A_2 != V_B_2 ) ) )).

fof(fact_fold__image__empty,axiom,(
    ! [V_z_2,V_g_2,V_f_2,T_c,T_b] : hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_b,T_c,V_f_2),V_g_2),V_z_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) = V_z_2 )).

fof(fact_folding__image__simple_Oeq__fold__g,axiom,(
    ! [V_A_2,V_F_2,V_g_2,V_z_2,V_f_2,T_c,T_b] :
      ( c_Finite__Set_Ofolding__image__simple(T_b,T_c,V_f_2,V_z_2,V_g_2,V_F_2)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2))
       => hAPP(V_F_2,V_A_2) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_b,T_c,V_f_2),V_g_2),V_z_2),V_A_2) ) ) )).

fof(fact_folding__one__idem_Oeq__fold__idem_H,axiom,(
    ! [V_a_2,V_A_2,V_F_2,V_f_2,T_b] :
      ( c_Finite__Set_Ofolding__one__idem(T_b,V_f_2,V_F_2)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => hAPP(V_F_2,hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2)) = hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,V_f_2),V_a_2),V_A_2) ) ) )).

fof(fact_fold1__eq__fold__idem,axiom,(
    ! [V_a_2,V_A_2,T_b] :
      ( class_Lattices_Oab__semigroup__idem__mult(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => hAPP(c_Finite__Set_Ofold1(T_b,c_Groups_Otimes__class_Otimes(T_b)),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2)) = hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Groups_Otimes__class_Otimes(T_b)),V_a_2),V_A_2) ) ) )).

fof(fact_folding__one_Oeq__fold_H,axiom,(
    ! [V_x_2,V_A_2,V_F_2,V_f_2,T_b] :
      ( c_Finite__Set_Ofolding__one(T_b,V_f_2,V_F_2)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( ~ hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
         => hAPP(V_F_2,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,V_f_2),V_x_2),V_A_2) ) ) ) )).

fof(fact_fun__left__comm_Ofold__rec,axiom,(
    ! [V_z_2,V_x_2,V_A_2,V_f_2,T_c,T_b] :
      ( c_Finite__Set_Ofun__left__comm(T_b,T_c,V_f_2)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
         => hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_c,V_f_2),V_z_2),V_A_2) = hAPP(hAPP(V_f_2,V_x_2),hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_c,V_f_2),V_z_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))) ) ) ) )).

fof(fact_comm__monoid__big_OF__eq,axiom,(
    ! [V_g_2,V_A_2,V_F_2,V_z_2,V_f_2,T_c,T_b] :
      ( c_Big__Operators_Ocomm__monoid__big(T_b,T_c,V_f_2,V_z_2,V_F_2)
     => ( ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2))
         => hAPP(hAPP(V_F_2,V_g_2),V_A_2) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_b,T_c,V_f_2),V_g_2),V_z_2),V_A_2) )
        & ( ~ hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2))
         => hAPP(hAPP(V_F_2,V_g_2),V_A_2) = V_z_2 ) ) ) )).

fof(fact_fun__left__comm_Ofold__insert__remove,axiom,(
    ! [V_x_2,V_z_2,V_A_2,V_f_2,T_c,T_b] :
      ( c_Finite__Set_Ofun__left__comm(T_b,T_c,V_f_2)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_c,V_f_2),V_z_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(V_f_2,V_x_2),hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_c,V_f_2),V_z_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))) ) ) )).

fof(fact_fold__graph_H_Oequations_I2_J,axiom,(
    ! [V_a5_2,V_a1_2,V_a2_2,V_a4_2,V_a3_2,T_c,T_b] :
      ( c_Nitpick_Ofold__graph_H(T_b,T_c,V_a3_2,V_a4_2,V_a2_2,hAPP(hAPP(V_a3_2,V_a1_2),V_a5_2))
    <=> ( ( V_a2_2 = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
          & hAPP(hAPP(V_a3_2,V_a1_2),V_a5_2) = V_a4_2 )
        | ? [B_x,B_y] :
            ( hAPP(hAPP(V_a3_2,V_a1_2),V_a5_2) = hAPP(hAPP(V_a3_2,B_x),B_y)
            & hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_a2_2))
            & c_Nitpick_Ofold__graph_H(T_b,T_c,V_a3_2,V_a4_2,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_a2_2),hAPP(hAPP(c_Set_Oinsert(T_b),B_x),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))),B_y) ) ) ) )).

fof(fact_fun__left__comm_Ofold__insert2,axiom,(
    ! [V_z_2,V_x_2,V_A_2,V_f_2,T_c,T_b] :
      ( c_Finite__Set_Ofun__left__comm(T_b,T_c,V_f_2)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( ~ hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
         => hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_c,V_f_2),V_z_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_c,V_f_2),hAPP(hAPP(V_f_2,V_x_2),V_z_2)),V_A_2) ) ) ) )).

fof(fact_fun__left__comm_Ofold__insert,axiom,(
    ! [V_z_2,V_x_2,V_A_2,V_f_2,T_c,T_b] :
      ( c_Finite__Set_Ofun__left__comm(T_b,T_c,V_f_2)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( ~ hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
         => hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_c,V_f_2),V_z_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(V_f_2,V_x_2),hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_c,V_f_2),V_z_2),V_A_2)) ) ) ) )).

fof(fact_fun__left__comm__idem_Ofold__insert__idem2,axiom,(
    ! [V_x_2,V_z_2,V_A_2,V_f_2,T_c,T_b] :
      ( c_Finite__Set_Ofun__left__comm__idem(T_b,T_c,V_f_2)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_c,V_f_2),V_z_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_c,V_f_2),hAPP(hAPP(V_f_2,V_x_2),V_z_2)),V_A_2) ) ) )).

fof(fact_fun__left__comm__idem_Ofold__insert__idem,axiom,(
    ! [V_x_2,V_z_2,V_A_2,V_f_2,T_c,T_b] :
      ( c_Finite__Set_Ofun__left__comm__idem(T_b,T_c,V_f_2)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_c,V_f_2),V_z_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(V_f_2,V_x_2),hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_c,V_f_2),V_z_2),V_A_2)) ) ) )).

fof(fact_setsum__diff1__ring,axiom,(
    ! [V_f_2,V_a_2,V_A_2,T_b,T_c] :
      ( class_Rings_Oring(T_c)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2))
         => hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),V_A_2)),hAPP(V_f_2,V_a_2)) ) ) ) )).

fof(fact_setsum__diff1,axiom,(
    ! [V_f_2,V_a_2,V_A_2,T_b,T_c] :
      ( class_Groups_Oab__group__add(T_c)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( ( hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2))
           => hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),V_A_2)),hAPP(V_f_2,V_a_2)) )
          & ( ~ hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2))
           => hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),V_A_2) ) ) ) ) )).

fof(fact_setprod_Oremove,axiom,(
    ! [V_g_2,V_x_2,V_A_2,T_b,T_c] :
      ( class_Groups_Ocomm__monoid__mult(T_c)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
         => hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_g_2),V_A_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(V_g_2,V_x_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_g_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))) ) ) ) )).

fof(fact_fun__left__comm__idem_Ofun__left__idem,axiom,(
    ! [V_z_2,V_x_2,V_f_2,T_c,T_b] :
      ( c_Finite__Set_Ofun__left__comm__idem(T_b,T_c,V_f_2)
     => hAPP(hAPP(V_f_2,V_x_2),hAPP(hAPP(V_f_2,V_x_2),V_z_2)) = hAPP(hAPP(V_f_2,V_x_2),V_z_2) ) )).

fof(fact_fun__left__comm_Ofun__left__comm,axiom,(
    ! [V_z_2,V_y_2,V_x_2,V_f_2,T_c,T_b] :
      ( c_Finite__Set_Ofun__left__comm(T_b,T_c,V_f_2)
     => hAPP(hAPP(V_f_2,V_x_2),hAPP(hAPP(V_f_2,V_y_2),V_z_2)) = hAPP(hAPP(V_f_2,V_y_2),hAPP(hAPP(V_f_2,V_x_2),V_z_2)) ) )).

fof(fact_fun__left__comm,axiom,(
    ! [T_b] :
      ( class_Groups_Oab__semigroup__mult(T_b)
     => c_Finite__Set_Ofun__left__comm(T_b,T_b,c_Groups_Otimes__class_Otimes(T_b)) ) )).

fof(fact_fun__left__comm__idem__insert,axiom,(
    ! [T_b] : c_Finite__Set_Ofun__left__comm__idem(T_b,tc_fun(T_b,tc_HOL_Obool),c_Set_Oinsert(T_b)) )).

fof(fact_fun__left__comm__idem,axiom,(
    ! [T_b] :
      ( class_Lattices_Oab__semigroup__idem__mult(T_b)
     => c_Finite__Set_Ofun__left__comm__idem(T_b,T_b,c_Groups_Otimes__class_Otimes(T_b)) ) )).

fof(fact_fun__left__comm_Ofold__graph__determ,axiom,(
    ! [V_y_2,V_x_2,V_A_2,V_z_2,V_f_2,T_c,T_b] :
      ( c_Finite__Set_Ofun__left__comm(T_b,T_c,V_f_2)
     => ( hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_b,T_c,V_f_2,V_z_2,V_A_2),V_x_2))
       => ( hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_b,T_c,V_f_2,V_z_2,V_A_2),V_y_2))
         => V_y_2 = V_x_2 ) ) ) )).

fof(fact_setsum__diff__nat,axiom,(
    ! [V_f_2,V_A_2,V_B_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2))
       => hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,tc_Nat_Onat),V_f_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,tc_Nat_Onat),V_f_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,tc_Nat_Onat),V_f_2),V_B_2)) ) ) )).

fof(fact_fun__left__comm_Ofold__fun__comm,axiom,(
    ! [V_z_2,V_x_2,V_A_2,V_f_2,T_c,T_b] :
      ( c_Finite__Set_Ofun__left__comm(T_b,T_c,V_f_2)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => hAPP(hAPP(V_f_2,V_x_2),hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_c,V_f_2),V_z_2),V_A_2)) = hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_c,V_f_2),hAPP(hAPP(V_f_2,V_x_2),V_z_2)),V_A_2) ) ) )).

fof(fact_setprod__insert,axiom,(
    ! [V_f_2,V_a_2,V_A_2,T_b,T_c] :
      ( class_Groups_Ocomm__monoid__mult(T_c)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( ~ hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2))
         => hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_f_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(V_f_2,V_a_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_f_2),V_A_2)) ) ) ) )).

fof(fact_setprod_Oinsert,axiom,(
    ! [V_g_2,V_x_2,V_A_2,T_b,T_c] :
      ( class_Groups_Ocomm__monoid__mult(T_c)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( ~ hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
         => hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_g_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(V_g_2,V_x_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_g_2),V_A_2)) ) ) ) )).

fof(fact_setsum__diff1__nat,axiom,(
    ! [V_f_2,V_A_2,V_a_2,T_b] :
      ( ( hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2))
       => hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,tc_Nat_Onat),V_f_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,tc_Nat_Onat),V_f_2),V_A_2)),hAPP(V_f_2,V_a_2)) )
      & ( ~ hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2))
       => hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,tc_Nat_Onat),V_f_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,tc_Nat_Onat),V_f_2),V_A_2) ) ) )).

fof(fact_setsum__diff,axiom,(
    ! [V_f_2,V_B_2,V_A_2,T_b,T_c] :
      ( class_Groups_Oab__group__add(T_c)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2))
         => hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),V_B_2)) ) ) ) )).

fof(fact_fun__left__comm_Ofold__equality,axiom,(
    ! [V_y_2,V_A_2,V_z_2,V_f_2,T_c,T_b] :
      ( c_Finite__Set_Ofun__left__comm(T_b,T_c,V_f_2)
     => ( hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_b,T_c,V_f_2,V_z_2,V_A_2),V_y_2))
       => hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_c,V_f_2),V_z_2),V_A_2) = V_y_2 ) ) )).

fof(fact_setprod_Oinsert__remove,axiom,(
    ! [V_x_2,V_g_2,V_A_2,T_b,T_c] :
      ( class_Groups_Ocomm__monoid__mult(T_c)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_g_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(V_g_2,V_x_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_g_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))) ) ) )).

fof(fact_comm__monoid__big_Oinfinite,axiom,(
    ! [V_g_2,V_A_2,V_F_2,V_z_2,V_f_2,T_c,T_b] :
      ( c_Big__Operators_Ocomm__monoid__big(T_b,T_c,V_f_2,V_z_2,V_F_2)
     => ( ~ hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2))
       => hAPP(hAPP(V_F_2,V_g_2),V_A_2) = V_z_2 ) ) )).

fof(fact_fun__left__comm_Ofold__graph__fold,axiom,(
    ! [V_z_2,V_A_2,V_f_2,T_c,T_b] :
      ( c_Finite__Set_Ofun__left__comm(T_b,T_c,V_f_2)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_b,T_c,V_f_2,V_z_2,V_A_2),hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_c,V_f_2),V_z_2),V_A_2))) ) ) )).

fof(fact_fun__left__comm_Ofold__graph__insertE__aux,axiom,(
    ! [V_a_2,V_y_2,V_A_2,V_z_2,V_f_2,T_c,T_b] :
      ( c_Finite__Set_Ofun__left__comm(T_b,T_c,V_f_2)
     => ( hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_b,T_c,V_f_2,V_z_2,V_A_2),V_y_2))
       => ( hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2))
         => ? [B_y_H] :
              ( V_y_2 = hAPP(hAPP(V_f_2,V_a_2),B_y_H)
              & hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_b,T_c,V_f_2,V_z_2,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))),B_y_H)) ) ) ) ) )).

fof(fact_setsum__strict__mono,axiom,(
    ! [V_g_2,V_f_2,V_A_2,T_b,T_c] :
      ( ( class_Groups_Ocomm__monoid__add(T_c)
        & class_Groups_Oordered__cancel__ab__semigroup__add(T_c) )
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
         => ( ! [B_x] :
                ( hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2))
               => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(V_f_2,B_x)),hAPP(V_g_2,B_x))) )
           => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_g_2),V_A_2))) ) ) ) ) )).

fof(fact_setsum__diff1_H,axiom,(
    ! [V_f_2,V_a_2,V_A_2,T_b,T_c] :
      ( class_Groups_Ocomm__monoid__add(T_c)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2))
         => hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),V_A_2) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(V_f_2,V_a_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))) ) ) ) )).

fof(fact_setsum_Oremove,axiom,(
    ! [V_g_2,V_x_2,V_A_2,T_b,T_c] :
      ( class_Groups_Ocomm__monoid__add(T_c)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
         => hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_g_2),V_A_2) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(V_g_2,V_x_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_g_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))) ) ) ) )).

fof(fact_fun__left__comm_Ofold__graph__insertE,axiom,(
    ! [V_v_2,V_A_2,V_x_2,V_z_2,V_f_2,T_c,T_b] :
      ( c_Finite__Set_Ofun__left__comm(T_b,T_c,V_f_2)
     => ( hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_b,T_c,V_f_2,V_z_2,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)),V_v_2))
       => ( ~ hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
         => ~ ! [B_y] :
                ( V_v_2 = hAPP(hAPP(V_f_2,V_x_2),B_y)
               => ~ hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_b,T_c,V_f_2,V_z_2,V_A_2),B_y)) ) ) ) ) )).

fof(fact_fold__image__cong,axiom,(
    ! [V_z_2,V_h_2,V_g_2,V_A_2,T_b,T_c] :
      ( class_Groups_Oab__semigroup__mult(T_c)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( ! [B_x] :
              ( hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2))
             => hAPP(V_g_2,B_x) = hAPP(V_h_2,B_x) )
         => hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_c,T_b,c_Groups_Otimes__class_Otimes(T_c)),V_g_2),V_z_2),V_A_2) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_c,T_b,c_Groups_Otimes__class_Otimes(T_c)),V_h_2),V_z_2),V_A_2) ) ) ) )).

fof(fact_setsum_Oinsert__remove,axiom,(
    ! [V_x_2,V_g_2,V_A_2,T_b,T_c] :
      ( class_Groups_Ocomm__monoid__add(T_c)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_g_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(V_g_2,V_x_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_g_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))) ) ) )).

fof(fact_fold1__antimono,axiom,(
    ! [V_B_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))
         => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2))
           => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Finite__Set_Ofold1(T_b,c_Orderings_Oord__class_Omin(T_b)),V_B_2)),hAPP(c_Finite__Set_Ofold1(T_b,c_Orderings_Oord__class_Omin(T_b)),V_A_2))) ) ) ) ) )).

fof(fact_card__Diff1__less,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
     => ( hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))),hAPP(c_Finite__Set_Ocard(T_b),V_A_2))) ) ) )).

fof(fact_card__Diff2__less,axiom,(
    ! [V_y_2,V_x_2,V_A_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
     => ( hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
       => ( hBOOL(hAPP(hAPP(c_member(T_b),V_y_2),V_A_2))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))),hAPP(hAPP(c_Set_Oinsert(T_b),V_y_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))),hAPP(c_Finite__Set_Ocard(T_b),V_A_2))) ) ) ) )).

fof(fact_Max__mono,axiom,(
    ! [V_N_2,V_M_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_M_2),V_N_2))
       => ( V_M_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
         => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_N_2))
           => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Big__Operators_Olinorder__class_OMax(T_b,V_M_2)),c_Big__Operators_Olinorder__class_OMax(T_b,V_N_2))) ) ) ) ) )).

fof(fact_add__right__imp__eq,axiom,(
    ! [V_c,V_a,V_b,T_a] :
      ( class_Groups_Ocancel__semigroup__add(T_a)
     => ( hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_a) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),V_a)
       => V_b = V_c ) ) )).

fof(fact_add__imp__eq,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Groups_Ocancel__ab__semigroup__add(T_a)
     => ( hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c)
       => V_b = V_c ) ) )).

fof(fact_add__left__imp__eq,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Groups_Ocancel__semigroup__add(T_a)
     => ( hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c)
       => V_b = V_c ) ) )).

fof(fact_add__right__cancel,axiom,(
    ! [V_c_2,V_a_2,V_b_2,T_b] :
      ( class_Groups_Ocancel__semigroup__add(T_b)
     => ( hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_b_2),V_a_2) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_c_2),V_a_2)
      <=> V_b_2 = V_c_2 ) ) )).

fof(fact_add__left__cancel,axiom,(
    ! [V_c_2,V_b_2,V_a_2,T_b] :
      ( class_Groups_Ocancel__semigroup__add(T_b)
     => ( hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_a_2),V_b_2) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_a_2),V_c_2)
      <=> V_b_2 = V_c_2 ) ) )).

fof(fact_min__add__distrib__left,axiom,(
    ! [V_z,V_y,V_x,T_a] :
      ( class_Groups_Oordered__ab__semigroup__add__imp__le(T_a)
     => hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),V_y)),V_z) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_x),V_z)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_y),V_z)) ) )).

fof(fact_min__max_Oinf__assoc,axiom,(
    ! [V_z,V_y,V_x,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),V_y)),V_z) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_y),V_z)) ) )).

fof(fact_min__max_Oinf_Oassoc,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_a),V_b)),V_c) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_b),V_c)) ) )).

fof(fact_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Groups_Oab__semigroup__add(T_a)
     => hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),V_c) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_c)) ) )).

fof(fact_min__max_Oinf__left__commute,axiom,(
    ! [V_z,V_y,V_x,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_y),V_z)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_y),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),V_z)) ) )).

fof(fact_min__max_Oinf_Oleft__commute,axiom,(
    ! [V_c,V_a,V_b,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_b),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_a),V_c)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_b),V_c)) ) )).

fof(fact_min__max_Oinf__left__idem,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),V_y)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),V_y) ) )).

fof(fact_min__max_Oinf_Oleft__idem,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_a),V_b)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_a),V_b) ) )).

fof(fact_min__max_Oinf__commute,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),V_y) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_y),V_x) ) )).

fof(fact_min__max_Oinf_Ocommute,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_a),V_b) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_b),V_a) ) )).

fof(fact_min__max_Oinf_Oidem,axiom,(
    ! [V_a,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_a),V_a) = V_a ) )).

fof(fact_Min_Oidem,axiom,(
    ! [V_x,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),V_x) = V_x ) )).

fof(fact_min__le__iff__disj,axiom,(
    ! [V_z_2,V_y_2,V_x_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),V_x_2),V_y_2)),V_z_2))
      <=> ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),V_z_2))
          | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_y_2),V_z_2)) ) ) ) )).

fof(fact_min__max_Oinf__le1,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),V_y)),V_x)) ) )).

fof(fact_min__max_Oinf__le2,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),V_y)),V_y)) ) )).

fof(fact_min__max_Ole__iff__inf,axiom,(
    ! [V_y_2,V_x_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),V_y_2))
      <=> hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),V_x_2),V_y_2) = V_x_2 ) ) )).

fof(fact_min__max_Ole__inf__iff,axiom,(
    ! [V_z_2,V_y_2,V_x_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),V_y_2),V_z_2)))
      <=> ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),V_y_2))
          & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),V_z_2)) ) ) ) )).

fof(fact_min__max_Ole__infI1,axiom,(
    ! [V_b,V_x,V_a,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_x))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_a),V_b)),V_x)) ) ) )).

fof(fact_min__max_Ole__infI2,axiom,(
    ! [V_a,V_x,V_b,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),V_x))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_a),V_b)),V_x)) ) ) )).

fof(fact_min__max_Oinf__absorb1,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y))
       => hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),V_y) = V_x ) ) )).

fof(fact_min__max_Oinf__absorb2,axiom,(
    ! [V_x,V_y,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),V_x))
       => hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),V_y) = V_y ) ) )).

fof(fact_min__max_Ole__infI,axiom,(
    ! [V_b,V_a,V_x,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_a))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_b))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_a),V_b))) ) ) ) )).

fof(fact_min__max_Oinf__greatest,axiom,(
    ! [V_z,V_y,V_x,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_z))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_y),V_z))) ) ) ) )).

fof(fact_min__max_Oinf__mono,axiom,(
    ! [V_d,V_b,V_c,V_a,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_c))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),V_d))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_a),V_b)),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_c),V_d))) ) ) ) )).

fof(fact_min__max_Ole__infE,axiom,(
    ! [V_b,V_a,V_x,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_a),V_b)))
       => ~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_a))
           => ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_b)) ) ) ) )).

fof(fact_min__max_Oless__infI2,axiom,(
    ! [V_a,V_x,V_b,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),V_x))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_a),V_b)),V_x)) ) ) )).

fof(fact_min__max_Oless__infI1,axiom,(
    ! [V_b,V_x,V_a,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_x))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_a),V_b)),V_x)) ) ) )).

fof(fact_min__less__iff__disj,axiom,(
    ! [V_z_2,V_y_2,V_x_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),V_x_2),V_y_2)),V_z_2))
      <=> ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_x_2),V_z_2))
          | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_y_2),V_z_2)) ) ) ) )).

fof(fact_min__less__iff__conj,axiom,(
    ! [V_y_2,V_x_2,V_z_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_z_2),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),V_x_2),V_y_2)))
      <=> ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_z_2),V_x_2))
          & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_z_2),V_y_2)) ) ) ) )).

fof(fact_min__diff__distrib__left,axiom,(
    ! [V_z,V_y,V_x,T_a] :
      ( class_Groups_Oordered__ab__group__add(T_a)
     => hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),V_y)),V_z) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_x),V_z)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_y),V_z)) ) )).

fof(fact_add__le__cancel__right,axiom,(
    ! [V_b_2,V_c_2,V_a_2,T_b] :
      ( class_Groups_Oordered__ab__semigroup__add__imp__le(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_a_2),V_c_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_b_2),V_c_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),V_b_2)) ) ) )).

fof(fact_add__le__cancel__left,axiom,(
    ! [V_b_2,V_a_2,V_c_2,T_b] :
      ( class_Groups_Oordered__ab__semigroup__add__imp__le(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_c_2),V_a_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_c_2),V_b_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),V_b_2)) ) ) )).

fof(fact_add__right__mono,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Groups_Oordered__ab__semigroup__add(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_c))) ) ) )).

fof(fact_add__left__mono,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Groups_Oordered__ab__semigroup__add(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),V_a)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),V_b))) ) ) )).

fof(fact_add__mono,axiom,(
    ! [V_d,V_c,V_b,V_a,T_a] :
      ( class_Groups_Oordered__ab__semigroup__add(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_c),V_d))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_d))) ) ) ) )).

fof(fact_add__le__imp__le__right,axiom,(
    ! [V_b,V_c,V_a,T_a] :
      ( class_Groups_Oordered__ab__semigroup__add__imp__le(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_c)))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b)) ) ) )).

fof(fact_add__le__imp__le__left,axiom,(
    ! [V_b,V_a,V_c,T_a] :
      ( class_Groups_Oordered__ab__semigroup__add__imp__le(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),V_a)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),V_b)))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b)) ) ) )).

fof(fact_add__less__imp__less__left,axiom,(
    ! [V_b,V_a,V_c,T_a] :
      ( class_Groups_Oordered__ab__semigroup__add__imp__le(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),V_a)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),V_b)))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)) ) ) )).

fof(fact_add__less__imp__less__right,axiom,(
    ! [V_b,V_c,V_a,T_a] :
      ( class_Groups_Oordered__ab__semigroup__add__imp__le(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_c)))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)) ) ) )).

fof(fact_add__strict__mono,axiom,(
    ! [V_d,V_c,V_b,V_a,T_a] :
      ( class_Groups_Oordered__cancel__ab__semigroup__add(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_c),V_d))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_d))) ) ) ) )).

fof(fact_add__strict__left__mono,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Groups_Oordered__cancel__ab__semigroup__add(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),V_a)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),V_b))) ) ) )).

fof(fact_add__strict__right__mono,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Groups_Oordered__cancel__ab__semigroup__add(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_c))) ) ) )).

fof(fact_add__less__cancel__left,axiom,(
    ! [V_b_2,V_a_2,V_c_2,T_b] :
      ( class_Groups_Oordered__ab__semigroup__add__imp__le(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_c_2),V_a_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_c_2),V_b_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),V_b_2)) ) ) )).

fof(fact_add__less__cancel__right,axiom,(
    ! [V_b_2,V_c_2,V_a_2,T_b] :
      ( class_Groups_Oordered__ab__semigroup__add__imp__le(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_a_2),V_c_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_b_2),V_c_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),V_b_2)) ) ) )).

fof(fact_diff__add__cancel,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Groups_Ogroup__add(T_a)
     => hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_b)),V_b) = V_a ) )).

fof(fact_add__diff__cancel,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Groups_Ogroup__add(T_a)
     => hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),V_b) = V_a ) )).

fof(fact_min__max_Ofun__left__comm__idem__inf,axiom,(
    ! [T_b] :
      ( class_Orderings_Olinorder(T_b)
     => c_Finite__Set_Ofun__left__comm__idem(T_b,T_b,c_Orderings_Oord__class_Omin(T_b)) ) )).

fof(fact_card__insert__le,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_b),V_A_2)),hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)))) ) )).

fof(fact_card__mono,axiom,(
    ! [V_A_2,V_B_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_b),V_A_2)),hAPP(c_Finite__Set_Ocard(T_b),V_B_2))) ) ) )).

fof(fact_card__seteq,axiom,(
    ! [V_A_2,V_B_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_b),V_B_2)),hAPP(c_Finite__Set_Ocard(T_b),V_A_2)))
         => V_A_2 = V_B_2 ) ) ) )).

fof(fact_diff__card__le__card__Diff,axiom,(
    ! [V_A_2,V_B_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_b),V_A_2)),hAPP(c_Finite__Set_Ocard(T_b),V_B_2))),hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)))) ) )).

fof(fact_add__less__le__mono,axiom,(
    ! [V_d,V_c,V_b,V_a,T_a] :
      ( class_Groups_Oordered__cancel__ab__semigroup__add(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_c),V_d))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_d))) ) ) ) )).

fof(fact_add__le__less__mono,axiom,(
    ! [V_d,V_c,V_b,V_a,T_a] :
      ( class_Groups_Oordered__cancel__ab__semigroup__add(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_c),V_d))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_d))) ) ) ) )).

fof(fact_psubset__card__mono,axiom,(
    ! [V_A_2,V_B_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_b),V_A_2)),hAPP(c_Finite__Set_Ocard(T_b),V_B_2))) ) ) )).

fof(fact_Max_Osingleton,axiom,(
    ! [V_x_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => c_Big__Operators_Olinorder__class_OMax(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) = V_x_2 ) )).

fof(fact_card__Diff__subset,axiom,(
    ! [V_A_2,V_B_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2))
       => hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_b),V_A_2)),hAPP(c_Finite__Set_Ocard(T_b),V_B_2)) ) ) )).

fof(fact_min__max_Ofold__inf__insert,axiom,(
    ! [V_a_2,V_b_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Orderings_Oord__class_Omin(T_b)),V_b_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),V_a_2),hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Orderings_Oord__class_Omin(T_b)),V_b_2),V_A_2)) ) ) )).

fof(fact_card__psubset,axiom,(
    ! [V_A_2,V_B_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_b),V_A_2)),hAPP(c_Finite__Set_Ocard(T_b),V_B_2)))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) ) ) ) )).

fof(fact_Max__ge,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),c_Big__Operators_Olinorder__class_OMax(T_b,V_A_2))) ) ) ) )).

fof(fact_Max__in,axiom,(
    ! [V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
         => hBOOL(hAPP(hAPP(c_member(T_b),c_Big__Operators_Olinorder__class_OMax(T_b,V_A_2)),V_A_2)) ) ) ) )).

fof(fact_card__Diff1__le,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))),hAPP(c_Finite__Set_Ocard(T_b),V_A_2))) ) )).

fof(fact_min__max_Ofold__inf__le__inf,axiom,(
    ! [V_b_2,V_a_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Orderings_Oord__class_Omin(T_b)),V_b_2),V_A_2)),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),V_a_2),V_b_2))) ) ) ) )).

fof(fact_min__max_Ofold1__belowI,axiom,(
    ! [V_a_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Finite__Set_Ofold1(T_b,c_Orderings_Oord__class_Omin(T_b)),V_A_2)),V_a_2)) ) ) ) )).

fof(fact_fold1__below__iff,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
         => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Finite__Set_Ofold1(T_b,c_Orderings_Oord__class_Omin(T_b)),V_A_2)),V_x_2))
          <=> ? [B_x] :
                ( hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2))
                & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),B_x),V_x_2)) ) ) ) ) ) )).

fof(fact_min__max_Obelow__fold1__iff,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
         => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),hAPP(c_Finite__Set_Ofold1(T_b,c_Orderings_Oord__class_Omin(T_b)),V_A_2)))
          <=> ! [B_x] :
                ( hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2))
               => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),B_x)) ) ) ) ) ) )).

fof(fact_fold1__strict__below__iff,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
         => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(c_Finite__Set_Ofold1(T_b,c_Orderings_Oord__class_Omin(T_b)),V_A_2)),V_x_2))
          <=> ? [B_x] :
                ( hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2))
                & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),B_x),V_x_2)) ) ) ) ) ) )).

fof(fact_strict__below__fold1__iff,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
         => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_x_2),hAPP(c_Finite__Set_Ofold1(T_b,c_Orderings_Oord__class_Omin(T_b)),V_A_2)))
          <=> ! [B_x] :
                ( hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2))
               => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_x_2),B_x)) ) ) ) ) ) )).

fof(fact_setsum__insert,axiom,(
    ! [V_f_2,V_a_2,V_F_2,T_b,T_c] :
      ( class_Groups_Ocomm__monoid__add(T_c)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_F_2))
       => ( ~ hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_F_2))
         => hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_F_2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(V_f_2,V_a_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),V_F_2)) ) ) ) )).

fof(fact_setsum_Oinsert,axiom,(
    ! [V_g_2,V_x_2,V_A_2,T_b,T_c] :
      ( class_Groups_Ocomm__monoid__add(T_c)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( ~ hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
         => hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_g_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(V_g_2,V_x_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_g_2),V_A_2)) ) ) ) )).

fof(fact_less__add__iff1,axiom,(
    ! [V_d_2,V_b_2,V_c_2,V_e_2,V_a_2,T_b] :
      ( class_Rings_Oordered__ring(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_e_2)),V_c_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_b_2),V_e_2)),V_d_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),V_a_2),V_b_2)),V_e_2)),V_c_2)),V_d_2)) ) ) )).

fof(fact_less__add__iff2,axiom,(
    ! [V_d_2,V_b_2,V_c_2,V_e_2,V_a_2,T_b] :
      ( class_Rings_Oordered__ring(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_e_2)),V_c_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_b_2),V_e_2)),V_d_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),V_b_2),V_a_2)),V_e_2)),V_d_2))) ) ) )).

fof(fact_le__add__iff1,axiom,(
    ! [V_d_2,V_b_2,V_c_2,V_e_2,V_a_2,T_b] :
      ( class_Rings_Oordered__ring(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_e_2)),V_c_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_b_2),V_e_2)),V_d_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),V_a_2),V_b_2)),V_e_2)),V_c_2)),V_d_2)) ) ) )).

fof(fact_le__add__iff2,axiom,(
    ! [V_d_2,V_b_2,V_c_2,V_e_2,V_a_2,T_b] :
      ( class_Rings_Oordered__ring(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_e_2)),V_c_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_b_2),V_e_2)),V_d_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_c_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),V_b_2),V_a_2)),V_e_2)),V_d_2))) ) ) )).

fof(fact_eq__add__iff1,axiom,(
    ! [V_d_2,V_b_2,V_c_2,V_e_2,V_a_2,T_b] :
      ( class_Rings_Oring(T_b)
     => ( hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_e_2)),V_c_2) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_b_2),V_e_2)),V_d_2)
      <=> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),V_a_2),V_b_2)),V_e_2)),V_c_2) = V_d_2 ) ) )).

fof(fact_eq__add__iff2,axiom,(
    ! [V_d_2,V_b_2,V_c_2,V_e_2,V_a_2,T_b] :
      ( class_Rings_Oring(T_b)
     => ( hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_e_2)),V_c_2) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_b_2),V_e_2)),V_d_2)
      <=> V_c_2 = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),V_b_2),V_a_2)),V_e_2)),V_d_2) ) ) )).

fof(fact_min__ord__min,axiom,(
    ! [T_b] :
      ( class_Orderings_Oord(T_b)
     => c_Orderings_Oord__class_Omin(T_b) = c_Orderings_Oord_Omin(T_b,c_Orderings_Oord__class_Oless__eq(T_b)) ) )).

fof(fact_card__Diff__singleton,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
     => ( hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
       => hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_b),V_A_2)),c_Groups_Oone__class_Oone(tc_Nat_Onat)) ) ) )).

fof(fact_card__Diff__singleton__if,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
     => ( ( hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
         => hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_b),V_A_2)),c_Groups_Oone__class_Oone(tc_Nat_Onat)) )
        & ( ~ hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
         => hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))) = hAPP(c_Finite__Set_Ocard(T_b),V_A_2) ) ) ) )).

fof(fact_less__diff__iff,axiom,(
    ! [V_n_2,V_m_2,V_k_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k_2),V_m_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k_2),V_n_2))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m_2),V_k_2)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n_2),V_k_2)))
        <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) ) ) ) )).

fof(fact_diff__less__mono,axiom,(
    ! [V_c,V_b,V_a] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_a),V_b))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_c),V_a))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_a),V_c)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_b),V_c))) ) ) )).

fof(fact_le__add2,axiom,(
    ! [V_m,V_n] : hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n))) )).

fof(fact_le__add1,axiom,(
    ! [V_m,V_n] : hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),V_m))) )).

fof(fact_le__iff__add,axiom,(
    ! [V_n_2,V_m_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2))
    <=> ? [B_k] : V_n_2 = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m_2),B_k) ) )).

fof(fact_nat__add__left__cancel__le,axiom,(
    ! [V_n_2,V_m_2,V_k_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k_2),V_m_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k_2),V_n_2)))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2)) ) )).

fof(fact_trans__le__add1,axiom,(
    ! [V_m,V_j,V_i] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i),V_j))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_j),V_m))) ) )).

fof(fact_trans__le__add2,axiom,(
    ! [V_m,V_j,V_i] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i),V_j))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_j))) ) )).

fof(fact_add__le__mono1,axiom,(
    ! [V_k,V_j,V_i] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i),V_j))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),V_k)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_j),V_k))) ) )).

fof(fact_add__le__mono,axiom,(
    ! [V_l,V_k,V_j,V_i] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i),V_j))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k),V_l))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),V_k)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_j),V_l))) ) ) )).

fof(fact_add__leD2,axiom,(
    ! [V_n,V_k,V_m] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_k)),V_n))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k),V_n)) ) )).

fof(fact_add__leD1,axiom,(
    ! [V_n,V_k,V_m] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_k)),V_n))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),V_n)) ) )).

fof(fact_add__leE,axiom,(
    ! [V_n,V_k,V_m] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_k)),V_n))
     => ~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),V_n))
         => ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k),V_n)) ) ) )).

fof(fact_le__antisym,axiom,(
    ! [V_n,V_m] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),V_n))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n),V_m))
       => V_m = V_n ) ) )).

fof(fact_le__trans,axiom,(
    ! [V_k,V_j,V_i] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i),V_j))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_j),V_k))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i),V_k)) ) ) )).

fof(fact_eq__imp__le,axiom,(
    ! [V_n,V_m] :
      ( V_m = V_n
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),V_n)) ) )).

fof(fact_nat__le__linear,axiom,(
    ! [V_n,V_m] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),V_n))
      | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n),V_m)) ) )).

fof(fact_le__refl,axiom,(
    ! [V_n] : hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n),V_n)) )).

fof(fact_nat__mult__1,axiom,(
    ! [V_n] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),V_n) = V_n )).

fof(fact_one__reorient,axiom,(
    ! [V_x_2,T_b] :
      ( class_Groups_Oone(T_b)
     => ( c_Groups_Oone__class_Oone(T_b) = V_x_2
      <=> V_x_2 = c_Groups_Oone__class_Oone(T_b) ) ) )).

fof(fact_nat__1__eq__mult__iff,axiom,(
    ! [V_n_2,V_m_2] :
      ( c_Groups_Oone__class_Oone(tc_Nat_Onat) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m_2),V_n_2)
    <=> ( V_m_2 = c_Groups_Oone__class_Oone(tc_Nat_Onat)
        & V_n_2 = c_Groups_Oone__class_Oone(tc_Nat_Onat) ) ) )).

fof(fact_nat__mult__1__right,axiom,(
    ! [V_n] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = V_n )).

fof(fact_nat__mult__eq__1__iff,axiom,(
    ! [V_n_2,V_m_2] :
      ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m_2),V_n_2) = c_Groups_Oone__class_Oone(tc_Nat_Onat)
    <=> ( V_m_2 = c_Groups_Oone__class_Oone(tc_Nat_Onat)
        & V_n_2 = c_Groups_Oone__class_Oone(tc_Nat_Onat) ) ) )).

fof(fact_le__square,axiom,(
    ! [V_m] : hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_m))) )).

fof(fact_le__cube,axiom,(
    ! [V_m] : hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_m)))) )).

fof(fact_mult__le__mono1,axiom,(
    ! [V_k,V_j,V_i] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i),V_j))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i),V_k)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j),V_k))) ) )).

fof(fact_mult__le__mono2,axiom,(
    ! [V_k,V_j,V_i] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i),V_j))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_i)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_j))) ) )).

fof(fact_mult__le__mono,axiom,(
    ! [V_l,V_k,V_j,V_i] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i),V_j))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k),V_l))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i),V_k)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j),V_l))) ) ) )).

fof(fact_mult_Ocomm__neutral,axiom,(
    ! [V_a,T_a] :
      ( class_Groups_Ocomm__monoid__mult(T_a)
     => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Oone__class_Oone(T_a)) = V_a ) )).

fof(fact_mult__1__right,axiom,(
    ! [V_a,T_a] :
      ( class_Groups_Omonoid__mult(T_a)
     => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Oone__class_Oone(T_a)) = V_a ) )).

fof(fact_mult__1,axiom,(
    ! [V_a,T_a] :
      ( class_Groups_Ocomm__monoid__mult(T_a)
     => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Oone__class_Oone(T_a)),V_a) = V_a ) )).

fof(fact_mult__1__left,axiom,(
    ! [V_a,T_a] :
      ( class_Groups_Omonoid__mult(T_a)
     => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Oone__class_Oone(T_a)),V_a) = V_a ) )).

fof(fact_setprod__eq__1__iff,axiom,(
    ! [V_f_2,V_F_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_F_2))
     => ( hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,tc_Nat_Onat),V_f_2),V_F_2) = c_Groups_Oone__class_Oone(tc_Nat_Onat)
      <=> ! [B_x] :
            ( hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_F_2))
           => hAPP(V_f_2,B_x) = c_Groups_Oone__class_Oone(tc_Nat_Onat) ) ) ) )).

fof(fact_less__add__one,axiom,(
    ! [V_a,T_a] :
      ( class_Rings_Olinordered__semidom(T_a)
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),c_Groups_Oone__class_Oone(T_a)))) ) )).

fof(fact_less__1__mult,axiom,(
    ! [V_n,V_m,T_a] :
      ( class_Rings_Olinordered__semidom(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Oone__class_Oone(T_a)),V_m))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Oone__class_Oone(T_a)),V_n))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Oone__class_Oone(T_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_m),V_n))) ) ) ) )).

fof(fact_setprod__empty,axiom,(
    ! [V_f_2,T_c,T_b] :
      ( class_Groups_Ocomm__monoid__mult(T_b)
     => hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_b),V_f_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) = c_Groups_Oone__class_Oone(T_b) ) )).

fof(fact_setprod_Oempty,axiom,(
    ! [V_g_2,T_c,T_b] :
      ( class_Groups_Ocomm__monoid__mult(T_b)
     => hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_b),V_g_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) = c_Groups_Oone__class_Oone(T_b) ) )).

fof(fact_setprod__infinite,axiom,(
    ! [V_f_2,V_A_2,T_b,T_c] :
      ( class_Groups_Ocomm__monoid__mult(T_c)
     => ( ~ hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_f_2),V_A_2) = c_Groups_Oone__class_Oone(T_c) ) ) )).

fof(fact_setprod_Oinfinite,axiom,(
    ! [V_g_2,V_A_2,T_b,T_c] :
      ( class_Groups_Ocomm__monoid__mult(T_c)
     => ( ~ hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_g_2),V_A_2) = c_Groups_Oone__class_Oone(T_c) ) ) )).

fof(fact_linorder__neqE__linordered__idom,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Rings_Olinordered__idom(T_a)
     => ( V_x != V_y
       => ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x)) ) ) ) )).

fof(fact_less__or__eq__imp__le,axiom,(
    ! [V_n,V_m] :
      ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n))
        | V_m = V_n )
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),V_n)) ) )).

fof(fact_le__neq__implies__less,axiom,(
    ! [V_n,V_m] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),V_n))
     => ( V_m != V_n
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) ) ) )).

fof(fact_less__imp__le__nat,axiom,(
    ! [V_n,V_m] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),V_n)) ) )).

fof(fact_le__eq__less__or__eq,axiom,(
    ! [V_n_2,V_m_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2))
    <=> ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2))
        | V_m_2 = V_n_2 ) ) )).

fof(fact_nat__less__le,axiom,(
    ! [V_n_2,V_m_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2))
    <=> ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2))
        & V_m_2 != V_n_2 ) ) )).

fof(fact_nat__less__cases,axiom,(
    ! [V_P_2,V_n_2,V_m_2] :
      ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2))
       => hBOOL(hAPP(hAPP(V_P_2,V_n_2),V_m_2)) )
     => ( ( V_m_2 = V_n_2
         => hBOOL(hAPP(hAPP(V_P_2,V_n_2),V_m_2)) )
       => ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),V_m_2))
           => hBOOL(hAPP(hAPP(V_P_2,V_n_2),V_m_2)) )
         => hBOOL(hAPP(hAPP(V_P_2,V_n_2),V_m_2)) ) ) ) )).

fof(fact_add__lessD1,axiom,(
    ! [V_k,V_j,V_i] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),V_j)),V_k))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),V_k)) ) )).

fof(fact_less__add__eq__less,axiom,(
    ! [V_n,V_m,V_l,V_k] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_k),V_l))
     => ( hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_l) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k),V_n)
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) ) ) )).

fof(fact_add__less__mono,axiom,(
    ! [V_l,V_k,V_j,V_i] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),V_j))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_k),V_l))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),V_k)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_j),V_l))) ) ) )).

fof(fact_add__less__mono1,axiom,(
    ! [V_k,V_j,V_i] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),V_j))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),V_k)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_j),V_k))) ) )).

fof(fact_trans__less__add2,axiom,(
    ! [V_m,V_j,V_i] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),V_j))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_j))) ) )).

fof(fact_trans__less__add1,axiom,(
    ! [V_m,V_j,V_i] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),V_j))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_j),V_m))) ) )).

fof(fact_less__not__refl3,axiom,(
    ! [V_t,V_s] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_s),V_t))
     => V_s != V_t ) )).

fof(fact_less__not__refl2,axiom,(
    ! [V_m,V_n] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),V_m))
     => V_m != V_n ) )).

fof(fact_less__irrefl__nat,axiom,(
    ! [V_n] : ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),V_n)) )).

fof(fact_linorder__neqE__nat,axiom,(
    ! [V_y,V_x] :
      ( V_x != V_y
     => ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_x),V_y))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_y),V_x)) ) ) )).

fof(fact_nat__add__left__cancel__less,axiom,(
    ! [V_n_2,V_m_2,V_k_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k_2),V_m_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k_2),V_n_2)))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) ) )).

fof(fact_nat__neq__iff,axiom,(
    ! [V_n_2,V_m_2] :
      ( V_m_2 != V_n_2
    <=> ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2))
        | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),V_m_2)) ) ) )).

fof(fact_not__add__less2,axiom,(
    ! [V_i,V_j] : ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_j),V_i)),V_i)) )).

fof(fact_not__add__less1,axiom,(
    ! [V_j,V_i] : ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),V_j)),V_i)) )).

fof(fact_less__not__refl,axiom,(
    ! [V_n] : ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),V_n)) )).

fof(fact_min__diff,axiom,(
    ! [V_n,V_i,V_m] : hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_i)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),V_i)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),V_m),V_n)),V_i) )).

fof(fact_diff__cancel2,axiom,(
    ! [V_n,V_k,V_m] : hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_k)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),V_k)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n) )).

fof(fact_diff__cancel,axiom,(
    ! [V_n,V_m,V_k] : hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k),V_m)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k),V_n)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n) )).

fof(fact_diff__mult__distrib,axiom,(
    ! [V_k,V_n,V_m] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)),V_k) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_k)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n),V_k)) )).

fof(fact_diff__commute,axiom,(
    ! [V_k,V_j,V_i] : hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_i),V_j)),V_k) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_i),V_k)),V_j) )).

fof(fact_diff__diff__left,axiom,(
    ! [V_k,V_j,V_i] : hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_i),V_j)),V_k) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_i),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_j),V_k)) )).

fof(fact_diff__add__inverse,axiom,(
    ! [V_m,V_n] : hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),V_m)),V_n) = V_m )).

fof(fact_diff__add__inverse2,axiom,(
    ! [V_n,V_m] : hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n)),V_n) = V_m )).

fof(fact_diff__mult__distrib2,axiom,(
    ! [V_n,V_m,V_k] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_m)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_n)) )).

fof(fact_diff__le__self,axiom,(
    ! [V_n,V_m] : hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)),V_m)) )).

fof(fact_diff__diff__right,axiom,(
    ! [V_i,V_j,V_k] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k),V_j))
     => hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_i),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j),V_k)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),V_k)),V_j) ) )).

fof(fact_le__diff__conv,axiom,(
    ! [V_i_2,V_k_2,V_j_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j_2),V_k_2)),V_i_2))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_j_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i_2),V_k_2))) ) )).

fof(fact_le__add__diff,axiom,(
    ! [V_m,V_n,V_k] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k),V_n))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),V_m)),V_k))) ) )).

fof(fact_diff__le__mono2,axiom,(
    ! [V_l,V_n,V_m] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),V_n))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_l),V_n)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_l),V_m))) ) )).

fof(fact_diff__le__mono,axiom,(
    ! [V_l,V_n,V_m] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),V_n))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_l)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),V_l))) ) )).

fof(fact_diff__diff__cancel,axiom,(
    ! [V_n,V_i] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i),V_n))
     => hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),V_i)) = V_i ) )).

fof(fact_le__add__diff__inverse,axiom,(
    ! [V_m,V_n] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n),V_m))
     => hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)) = V_m ) )).

fof(fact_add__diff__assoc,axiom,(
    ! [V_i,V_j,V_k] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k),V_j))
     => hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j),V_k)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),V_j)),V_k) ) )).

fof(fact_le__diff__conv2,axiom,(
    ! [V_i_2,V_j_2,V_k_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k_2),V_j_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j_2),V_k_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i_2),V_k_2)),V_j_2)) ) ) )).

fof(fact_le__add__diff__inverse2,axiom,(
    ! [V_m,V_n] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n),V_m))
     => hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)),V_n) = V_m ) )).

fof(fact_le__imp__diff__is__add,axiom,(
    ! [V_k_2,V_j_2,V_i_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i_2),V_j_2))
     => ( hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j_2),V_i_2) = V_k_2
      <=> V_j_2 = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k_2),V_i_2) ) ) )).

fof(fact_diff__add__assoc,axiom,(
    ! [V_i,V_j,V_k] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k),V_j))
     => hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),V_j)),V_k) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j),V_k)) ) )).

fof(fact_add__diff__assoc2,axiom,(
    ! [V_i,V_j,V_k] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k),V_j))
     => hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j),V_k)),V_i) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_j),V_i)),V_k) ) )).

fof(fact_diff__add__assoc2,axiom,(
    ! [V_i,V_j,V_k] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k),V_j))
     => hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_j),V_i)),V_k) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j),V_k)),V_i) ) )).

fof(fact_eq__diff__iff,axiom,(
    ! [V_n_2,V_m_2,V_k_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k_2),V_m_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k_2),V_n_2))
       => ( hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m_2),V_k_2) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n_2),V_k_2)
        <=> V_m_2 = V_n_2 ) ) ) )).

fof(fact_Nat_Odiff__diff__eq,axiom,(
    ! [V_n,V_m,V_k] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k),V_m))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k),V_n))
       => hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_k)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),V_k)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n) ) ) )).

fof(fact_le__diff__iff,axiom,(
    ! [V_n_2,V_m_2,V_k_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k_2),V_m_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k_2),V_n_2))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m_2),V_k_2)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n_2),V_k_2)))
        <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2)) ) ) ) )).

fof(fact_card_Oinsert,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
     => ( ~ hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
       => hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(c_Finite__Set_Ocard(T_b),V_A_2)) ) ) )).

fof(fact_setprod_Oeq__fold,axiom,(
    ! [V_g_2,V_A_2,T_b,T_c] :
      ( class_Groups_Ocomm__monoid__mult(T_c)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_g_2),V_A_2) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_c,T_b,c_Groups_Otimes__class_Otimes(T_c)),V_g_2),c_Groups_Oone__class_Oone(T_c)),V_A_2) ) ) )).

fof(fact_setprod_OF__eq,axiom,(
    ! [V_g_2,V_A_2,T_b,T_c] :
      ( class_Groups_Ocomm__monoid__mult(T_c)
     => ( ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
         => hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_g_2),V_A_2) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_c,T_b,c_Groups_Otimes__class_Otimes(T_c)),V_g_2),c_Groups_Oone__class_Oone(T_c)),V_A_2) )
        & ( ~ hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
         => hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_g_2),V_A_2) = c_Groups_Oone__class_Oone(T_c) ) ) ) )).

fof(fact_card_Oinsert__remove,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
     => hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))) ) )).

fof(fact_card_Oremove,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
     => ( hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
       => hAPP(c_Finite__Set_Ocard(T_b),V_A_2) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))) ) ) )).

fof(fact_card__Diff__insert,axiom,(
    ! [V_B_2,V_a_2,V_A_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
     => ( hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2))
       => ( ~ hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_B_2))
         => hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_B_2))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))),c_Groups_Oone__class_Oone(tc_Nat_Onat)) ) ) ) )).

fof(fact_combine__common__factor,axiom,(
    ! [V_c,V_b,V_e,V_a,T_a] :
      ( class_Rings_Osemiring(T_a)
     => hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_e)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_e)),V_c)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),V_e)),V_c) ) )).

fof(fact_comm__semiring__class_Odistrib,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Rings_Ocomm__semiring(T_a)
     => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),V_c) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c)) ) )).

fof(fact_diff__less__mono2,axiom,(
    ! [V_l,V_n,V_m] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_l))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_l),V_n)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_l),V_m))) ) ) )).

fof(fact_less__imp__diff__less,axiom,(
    ! [V_n,V_k,V_j] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_j),V_k))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j),V_n)),V_k)) ) )).

fof(fact_add__diff__inverse,axiom,(
    ! [V_n,V_m] :
      ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n))
     => hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)) = V_m ) )).

fof(fact_less__diff__conv,axiom,(
    ! [V_k_2,V_j_2,V_i_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j_2),V_k_2)))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i_2),V_k_2)),V_j_2)) ) )).

fof(fact_nat__less__add__iff2,axiom,(
    ! [V_n_2,V_m_2,V_u_2,V_j_2,V_i_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i_2),V_j_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i_2),V_u_2)),V_m_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j_2),V_u_2)),V_n_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j_2),V_i_2)),V_u_2)),V_n_2))) ) ) )).

fof(fact_nat__less__add__iff1,axiom,(
    ! [V_n_2,V_m_2,V_u_2,V_i_2,V_j_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_j_2),V_i_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i_2),V_u_2)),V_m_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j_2),V_u_2)),V_n_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_i_2),V_j_2)),V_u_2)),V_m_2)),V_n_2)) ) ) )).

fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,(
    ! [V_m,T_a] :
      ( class_Rings_Ocomm__semiring__1(T_a)
     => hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_m),V_m) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Groups_Oone__class_Oone(T_a)),c_Groups_Oone__class_Oone(T_a))),V_m) ) )).

fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,(
    ! [V_a,V_m,T_a] :
      ( class_Rings_Ocomm__semiring__1(T_a)
     => hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_m),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_m)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),c_Groups_Oone__class_Oone(T_a))),V_m) ) )).

fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,(
    ! [V_m,V_a,T_a] :
      ( class_Rings_Ocomm__semiring__1(T_a)
     => hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_m)),V_m) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),c_Groups_Oone__class_Oone(T_a))),V_m) ) )).

fof(fact_nat__eq__add__iff2,axiom,(
    ! [V_n_2,V_m_2,V_u_2,V_j_2,V_i_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i_2),V_j_2))
     => ( hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i_2),V_u_2)),V_m_2) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j_2),V_u_2)),V_n_2)
      <=> V_m_2 = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j_2),V_i_2)),V_u_2)),V_n_2) ) ) )).

fof(fact_nat__diff__add__eq2,axiom,(
    ! [V_n,V_m,V_u,V_j,V_i] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i),V_j))
     => hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i),V_u)),V_m)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j),V_u)),V_n)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j),V_i)),V_u)),V_n)) ) )).

fof(fact_nat__le__add__iff2,axiom,(
    ! [V_n_2,V_m_2,V_u_2,V_j_2,V_i_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i_2),V_j_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i_2),V_u_2)),V_m_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j_2),V_u_2)),V_n_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j_2),V_i_2)),V_u_2)),V_n_2))) ) ) )).

fof(fact_nat__eq__add__iff1,axiom,(
    ! [V_n_2,V_m_2,V_u_2,V_i_2,V_j_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_j_2),V_i_2))
     => ( hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i_2),V_u_2)),V_m_2) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j_2),V_u_2)),V_n_2)
      <=> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_i_2),V_j_2)),V_u_2)),V_m_2) = V_n_2 ) ) )).

fof(fact_nat__diff__add__eq1,axiom,(
    ! [V_n,V_m,V_u,V_i,V_j] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_j),V_i))
     => hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i),V_u)),V_m)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j),V_u)),V_n)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_i),V_j)),V_u)),V_m)),V_n) ) )).

fof(fact_add__mult__distrib,axiom,(
    ! [V_k,V_n,V_m] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n)),V_k) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_k)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n),V_k)) )).

fof(fact_add__mult__distrib2,axiom,(
    ! [V_n,V_m,V_k] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_m)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_n)) )).

fof(fact_nat__add__commute,axiom,(
    ! [V_n,V_m] : hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),V_m) )).

fof(fact_nat__add__left__commute,axiom,(
    ! [V_z,V_y,V_x] : hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_x),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_y),V_z)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_y),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_x),V_z)) )).

fof(fact_nat__add__assoc,axiom,(
    ! [V_k,V_n,V_m] : hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n)),V_k) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),V_k)) )).

fof(fact_nat__add__left__cancel,axiom,(
    ! [V_n_2,V_m_2,V_k_2] :
      ( hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k_2),V_m_2) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k_2),V_n_2)
    <=> V_m_2 = V_n_2 ) )).

fof(fact_nat__add__right__cancel,axiom,(
    ! [V_n_2,V_k_2,V_m_2] :
      ( hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m_2),V_k_2) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n_2),V_k_2)
    <=> V_m_2 = V_n_2 ) )).

fof(fact_nat__mult__commute,axiom,(
    ! [V_n,V_m] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n),V_m) )).

fof(fact_nat__mult__assoc,axiom,(
    ! [V_k,V_n,V_m] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n)),V_k) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n),V_k)) )).

fof(fact_left__add__mult__distrib,axiom,(
    ! [V_k,V_j,V_u,V_i] : hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i),V_u)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j),V_u)),V_k)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),V_j)),V_u)),V_k) )).

fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,(
    ! [V_c,V_a,T_a] :
      ( class_Rings_Ocomm__semiring__1(T_a)
     => hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),V_a) ) )).

fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,(
    ! [V_d,V_c,V_a,T_a] :
      ( class_Rings_Ocomm__semiring__1(T_a)
     => hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),V_d)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_d)) ) )).

fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,(
    ! [V_d,V_c,V_a,T_a] :
      ( class_Rings_Ocomm__semiring__1(T_a)
     => hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),V_d)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c)),V_d) ) )).

fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Rings_Ocomm__semiring__1(T_a)
     => hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),V_c) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_c)) ) )).

fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Rings_Ocomm__semiring__1(T_a)
     => hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),V_c) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c)),V_b) ) )).

fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,(
    ! [V_d,V_c,V_b,V_a,T_a] :
      ( class_Rings_Ocomm__semiring__1(T_a)
     => hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),V_d)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_d)) ) )).

fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Rings_Ocomm__semiring__1(T_a)
     => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a) ) )).

fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,(
    ! [V_ry,V_rx,V_lx,T_a] :
      ( class_Rings_Ocomm__semiring__1(T_a)
     => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_rx),V_ry)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_rx),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),V_ry)) ) )).

fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,(
    ! [V_ry,V_rx,V_lx,T_a] :
      ( class_Rings_Ocomm__semiring__1(T_a)
     => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_rx),V_ry)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),V_rx)),V_ry) ) )).

fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,(
    ! [V_rx,V_ly,V_lx,T_a] :
      ( class_Rings_Ocomm__semiring__1(T_a)
     => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),V_ly)),V_rx) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_ly),V_rx)) ) )).

fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,(
    ! [V_rx,V_ly,V_lx,T_a] :
      ( class_Rings_Ocomm__semiring__1(T_a)
     => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),V_ly)),V_rx) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),V_rx)),V_ly) ) )).

fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,(
    ! [V_ry,V_rx,V_ly,V_lx,T_a] :
      ( class_Rings_Ocomm__semiring__1(T_a)
     => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),V_ly)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_rx),V_ry)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_ly),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_rx),V_ry))) ) )).

fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,(
    ! [V_ry,V_rx,V_ly,V_lx,T_a] :
      ( class_Rings_Ocomm__semiring__1(T_a)
     => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),V_ly)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_rx),V_ry)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_rx),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),V_ly)),V_ry)) ) )).

fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,(
    ! [V_ry,V_rx,V_ly,V_lx,T_a] :
      ( class_Rings_Ocomm__semiring__1(T_a)
     => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),V_ly)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_rx),V_ry)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),V_rx)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_ly),V_ry)) ) )).

fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,(
    ! [V_z,V_y,V_x,T_a] :
      ( class_Rings_Ocomm__semiring__1(T_a)
     => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_y),V_z)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_y)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_z)) ) )).

fof(fact_crossproduct__noteq,axiom,(
    ! [V_d_2,V_c_2,V_b_2,V_a_2,T_b] :
      ( class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(T_b)
     => ( ( V_a_2 != V_b_2
          & V_c_2 != V_d_2 )
      <=> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_c_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_b_2),V_d_2)) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_d_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_b_2),V_c_2)) ) ) )).

fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Rings_Ocomm__semiring__1(T_a)
     => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),V_c) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c)) ) )).

fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,(
    ! [V_b,V_m,V_a,T_a] :
      ( class_Rings_Ocomm__semiring__1(T_a)
     => hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_m)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_m)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),V_m) ) )).

fof(fact_crossproduct__eq,axiom,(
    ! [V_z_2,V_x_2,V_y_2,V_w_2,T_b] :
      ( class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(T_b)
     => ( hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_w_2),V_y_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_x_2),V_z_2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_w_2),V_z_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_x_2),V_y_2))
      <=> ( V_w_2 = V_x_2
          | V_y_2 = V_z_2 ) ) ) )).

fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,(
    ! [V_a,T_a] :
      ( class_Rings_Ocomm__semiring__1(T_a)
     => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Oone__class_Oone(T_a)),V_a) = V_a ) )).

fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,(
    ! [V_a,T_a] :
      ( class_Rings_Ocomm__semiring__1(T_a)
     => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Oone__class_Oone(T_a)) = V_a ) )).

fof(fact_nat__le__add__iff1,axiom,(
    ! [V_n_2,V_m_2,V_u_2,V_i_2,V_j_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_j_2),V_i_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i_2),V_u_2)),V_m_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j_2),V_u_2)),V_n_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_i_2),V_j_2)),V_u_2)),V_m_2)),V_n_2)) ) ) )).

fof(fact_termination__basic__simps_I4_J,axiom,(
    ! [V_y,V_z,V_x] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_x),V_z))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_x),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_y),V_z))) ) )).

fof(fact_termination__basic__simps_I3_J,axiom,(
    ! [V_z,V_y,V_x] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_x),V_y))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_x),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_y),V_z))) ) )).

fof(fact_termination__basic__simps_I5_J,axiom,(
    ! [V_y,V_x] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_x),V_y))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_x),V_y)) ) )).

fof(fact_termination__basic__simps_I1_J,axiom,(
    ! [V_z,V_y,V_x] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_x),V_y))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_x),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_y),V_z))) ) )).

fof(fact_termination__basic__simps_I2_J,axiom,(
    ! [V_y,V_z,V_x] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_x),V_z))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_x),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_y),V_z))) ) )).

fof(fact_Min_Oremove,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
         => ( ( hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
             => c_Big__Operators_Olinorder__class_OMin(T_b,V_A_2) = V_x_2 )
            & ( hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
             => c_Big__Operators_Olinorder__class_OMin(T_b,V_A_2) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),V_x_2),c_Big__Operators_Olinorder__class_OMin(T_b,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))) ) ) ) ) ) )).

fof(fact_min__max_OInf__fin_Oremove,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
         => ( ( hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
             => c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),V_A_2) = V_x_2 )
            & ( hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
             => c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),V_A_2) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),V_x_2),c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))) ) ) ) ) ) )).

fof(fact_setprod__mono__one__right,axiom,(
    ! [V_f_2,V_S_2,V_T_2,T_b,T_c] :
      ( class_Groups_Ocomm__monoid__mult(T_c)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_T_2))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_S_2),V_T_2))
         => ( ! [B_x] :
                ( hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_T_2),V_S_2)))
               => hAPP(V_f_2,B_x) = c_Groups_Oone__class_Oone(T_c) )
           => hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_f_2),V_T_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_f_2),V_S_2) ) ) ) ) )).

fof(fact_setprod__mono__one__left,axiom,(
    ! [V_f_2,V_S_2,V_T_2,T_b,T_c] :
      ( class_Groups_Ocomm__monoid__mult(T_c)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_T_2))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_S_2),V_T_2))
         => ( ! [B_x] :
                ( hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_T_2),V_S_2)))
               => hAPP(V_f_2,B_x) = c_Groups_Oone__class_Oone(T_c) )
           => hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_f_2),V_S_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_f_2),V_T_2) ) ) ) ) )).

fof(fact_card__Suc__Diff1,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
     => ( hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
       => hAPP(c_Nat_OSuc,hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))) = hAPP(c_Finite__Set_Ocard(T_b),V_A_2) ) ) )).

fof(fact_Suc__mono,axiom,(
    ! [V_n,V_m] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),hAPP(c_Nat_OSuc,V_n))) ) )).

fof(fact_lessI,axiom,(
    ! [V_n] : hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),hAPP(c_Nat_OSuc,V_n))) )).

fof(fact_n__not__Suc__n,axiom,(
    ! [V_n] : V_n != hAPP(c_Nat_OSuc,V_n) )).

fof(fact_Suc__n__not__n,axiom,(
    ! [V_n] : hAPP(c_Nat_OSuc,V_n) != V_n )).

fof(fact_nat_Oinject,axiom,(
    ! [V_nat_H_2,V_nat_2] :
      ( hAPP(c_Nat_OSuc,V_nat_2) = hAPP(c_Nat_OSuc,V_nat_H_2)
    <=> V_nat_2 = V_nat_H_2 ) )).

fof(fact_Suc__inject,axiom,(
    ! [V_y,V_x] :
      ( hAPP(c_Nat_OSuc,V_x) = hAPP(c_Nat_OSuc,V_y)
     => V_x = V_y ) )).

fof(fact_Suc__less__SucD,axiom,(
    ! [V_n,V_m] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),hAPP(c_Nat_OSuc,V_n)))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) ) )).

fof(fact_Suc__lessD,axiom,(
    ! [V_n,V_m] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),V_n))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) ) )).

fof(fact_less__SucE,axiom,(
    ! [V_n,V_m] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),hAPP(c_Nat_OSuc,V_n)))
     => ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n))
       => V_m = V_n ) ) )).

fof(fact_less__trans__Suc,axiom,(
    ! [V_k,V_j,V_i] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),V_j))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_j),V_k))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_i)),V_k)) ) ) )).

fof(fact_Suc__lessI,axiom,(
    ! [V_n,V_m] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n))
     => ( hAPP(c_Nat_OSuc,V_m) != V_n
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),V_n)) ) ) )).

fof(fact_less__SucI,axiom,(
    ! [V_n,V_m] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),hAPP(c_Nat_OSuc,V_n))) ) )).

fof(fact_less__antisym,axiom,(
    ! [V_m,V_n] :
      ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),V_m))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),hAPP(c_Nat_OSuc,V_m)))
       => V_m = V_n ) ) )).

fof(fact_not__less__less__Suc__eq,axiom,(
    ! [V_m_2,V_n_2] :
      ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),V_m_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),hAPP(c_Nat_OSuc,V_m_2)))
      <=> V_n_2 = V_m_2 ) ) )).

fof(fact_Suc__less__eq,axiom,(
    ! [V_n_2,V_m_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m_2)),hAPP(c_Nat_OSuc,V_n_2)))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) ) )).

fof(fact_less__Suc__eq,axiom,(
    ! [V_n_2,V_m_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),hAPP(c_Nat_OSuc,V_n_2)))
    <=> ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2))
        | V_m_2 = V_n_2 ) ) )).

fof(fact_not__less__eq,axiom,(
    ! [V_n_2,V_m_2] :
      ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),hAPP(c_Nat_OSuc,V_m_2))) ) )).

fof(fact_add__Suc__shift,axiom,(
    ! [V_n,V_m] : hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),V_n) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),hAPP(c_Nat_OSuc,V_n)) )).

fof(fact_add__Suc,axiom,(
    ! [V_n,V_m] : hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),V_n) = hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n)) )).

fof(fact_add__Suc__right,axiom,(
    ! [V_n,V_m] : hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),hAPP(c_Nat_OSuc,V_n)) = hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n)) )).

fof(fact_Suc__leD,axiom,(
    ! [V_n,V_m] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),V_n))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),V_n)) ) )).

fof(fact_le__SucE,axiom,(
    ! [V_n,V_m] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),hAPP(c_Nat_OSuc,V_n)))
     => ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),V_n))
       => V_m = hAPP(c_Nat_OSuc,V_n) ) ) )).

fof(fact_le__SucI,axiom,(
    ! [V_n,V_m] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),V_n))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),hAPP(c_Nat_OSuc,V_n))) ) )).

fof(fact_Suc__le__mono,axiom,(
    ! [V_m_2,V_n_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_n_2)),hAPP(c_Nat_OSuc,V_m_2)))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n_2),V_m_2)) ) )).

fof(fact_le__Suc__eq,axiom,(
    ! [V_n_2,V_m_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),hAPP(c_Nat_OSuc,V_n_2)))
    <=> ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2))
        | V_m_2 = hAPP(c_Nat_OSuc,V_n_2) ) ) )).

fof(fact_not__less__eq__eq,axiom,(
    ! [V_n_2,V_m_2] :
      ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_n_2)),V_m_2)) ) )).

fof(fact_Suc__n__not__le__n,axiom,(
    ! [V_n] : ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_n)),V_n)) )).

fof(fact_Suc__diff__diff,axiom,(
    ! [V_k,V_n,V_m] : hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),V_n)),hAPP(c_Nat_OSuc,V_k)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)),V_k) )).

fof(fact_diff__Suc__Suc,axiom,(
    ! [V_n,V_m] : hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),hAPP(c_Nat_OSuc,V_n)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n) )).

fof(fact_Suc__mult__cancel1,axiom,(
    ! [V_n_2,V_m_2,V_k_2] :
      ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_k_2)),V_m_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_k_2)),V_n_2)
    <=> V_m_2 = V_n_2 ) )).

fof(fact_min__Suc__Suc,axiom,(
    ! [V_n,V_m] : hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),hAPP(c_Nat_OSuc,V_n)) = hAPP(c_Nat_OSuc,hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),V_m),V_n)) )).

fof(fact_less__iff__Suc__add,axiom,(
    ! [V_n_2,V_m_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2))
    <=> ? [B_k] : V_n_2 = hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m_2),B_k)) ) )).

fof(fact_less__add__Suc2,axiom,(
    ! [V_m,V_i] : hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_i)))) )).

fof(fact_less__add__Suc1,axiom,(
    ! [V_m,V_i] : hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),V_m)))) )).

fof(fact_Suc__le__lessD,axiom,(
    ! [V_n,V_m] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),V_n))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) ) )).

fof(fact_le__less__Suc__eq,axiom,(
    ! [V_n_2,V_m_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),hAPP(c_Nat_OSuc,V_m_2)))
      <=> V_n_2 = V_m_2 ) ) )).

fof(fact_Suc__leI,axiom,(
    ! [V_n,V_m] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),V_n)) ) )).

fof(fact_le__imp__less__Suc,axiom,(
    ! [V_n,V_m] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),V_n))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),hAPP(c_Nat_OSuc,V_n))) ) )).

fof(fact_Suc__le__eq,axiom,(
    ! [V_n_2,V_m_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m_2)),V_n_2))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) ) )).

fof(fact_less__Suc__eq__le,axiom,(
    ! [V_n_2,V_m_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),hAPP(c_Nat_OSuc,V_n_2)))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2)) ) )).

fof(fact_less__eq__Suc__le,axiom,(
    ! [V_m_2,V_n_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),V_m_2))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_n_2)),V_m_2)) ) )).

fof(fact_diff__less__Suc,axiom,(
    ! [V_n,V_m] : hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)),hAPP(c_Nat_OSuc,V_m))) )).

fof(fact_Suc__mult__less__cancel1,axiom,(
    ! [V_n_2,V_m_2,V_k_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_k_2)),V_m_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_k_2)),V_n_2)))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) ) )).

fof(fact_mult__Suc__right,axiom,(
    ! [V_n,V_m] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),hAPP(c_Nat_OSuc,V_n)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n)) )).

fof(fact_mult__Suc,axiom,(
    ! [V_n,V_m] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),V_n) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n)) )).

fof(fact_Suc__diff__le,axiom,(
    ! [V_m,V_n] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n),V_m))
     => hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),V_n) = hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)) ) )).

fof(fact_Suc__mult__le__cancel1,axiom,(
    ! [V_n_2,V_m_2,V_k_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_k_2)),V_m_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_k_2)),V_n_2)))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2)) ) )).

fof(fact_diff__Suc__1,axiom,(
    ! [V_n] : hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_n)),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = V_n )).

fof(fact_Min_Osingleton,axiom,(
    ! [V_x_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => c_Big__Operators_Olinorder__class_OMin(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) = V_x_2 ) )).

fof(fact_diff__Suc__diff__eq1,axiom,(
    ! [V_m,V_j,V_k] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k),V_j))
     => hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j),V_k))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_k)),hAPP(c_Nat_OSuc,V_j)) ) )).

fof(fact_diff__Suc__diff__eq2,axiom,(
    ! [V_m,V_j,V_k] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k),V_j))
     => hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j),V_k))),V_m) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_j)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k),V_m)) ) )).

fof(fact_min__max_OInf__fin_Oin__idem,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
         => hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),V_x_2),c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),V_A_2)) = c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),V_A_2) ) ) ) )).

fof(fact_Min__le,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Big__Operators_Olinorder__class_OMin(T_b,V_A_2)),V_x_2)) ) ) ) )).

fof(fact_min__max_OInf__fin_Osingleton,axiom,(
    ! [V_x_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) = V_x_2 ) )).

fof(fact_Min__in,axiom,(
    ! [V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
         => hBOOL(hAPP(hAPP(c_member(T_b),c_Big__Operators_Olinorder__class_OMin(T_b,V_A_2)),V_A_2)) ) ) ) )).

fof(fact_Min_Oin__idem,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
         => hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),V_x_2),c_Big__Operators_Olinorder__class_OMin(T_b,V_A_2)) = c_Big__Operators_Olinorder__class_OMin(T_b,V_A_2) ) ) ) )).

fof(fact_min__max_OInf__fin_OF__eq,axiom,(
    ! [V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),V_A_2) = hAPP(c_Finite__Set_Ofold1(T_b,c_Orderings_Oord__class_Omin(T_b)),V_A_2) ) ) )).

fof(fact_Min_OF__eq,axiom,(
    ! [V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => c_Big__Operators_Olinorder__class_OMin(T_b,V_A_2) = hAPP(c_Finite__Set_Ofold1(T_b,c_Orderings_Oord__class_Omin(T_b)),V_A_2) ) ) )).

fof(fact_card__insert__if,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
     => ( ( hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
         => hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(c_Finite__Set_Ocard(T_b),V_A_2) )
        & ( ~ hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
         => hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(c_Nat_OSuc,hAPP(c_Finite__Set_Ocard(T_b),V_A_2)) ) ) ) )).

fof(fact_card__insert__disjoint,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
     => ( ~ hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
       => hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(c_Nat_OSuc,hAPP(c_Finite__Set_Ocard(T_b),V_A_2)) ) ) )).

fof(fact_min__max_OInf__fin_Oinsert__idem,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
         => c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),V_x_2),c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),V_A_2)) ) ) ) )).

fof(fact_min__max_OInf__fin_Osubset__idem,axiom,(
    ! [V_B_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( V_B_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
         => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2))
           => hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),V_B_2)),c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),V_A_2)) = c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),V_A_2) ) ) ) ) )).

fof(fact_Min__antimono,axiom,(
    ! [V_N_2,V_M_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_M_2),V_N_2))
       => ( V_M_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
         => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_N_2))
           => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Big__Operators_Olinorder__class_OMin(T_b,V_N_2)),c_Big__Operators_Olinorder__class_OMin(T_b,V_M_2))) ) ) ) ) )).

fof(fact_min__max_OInf__fin_Oeq__fold__idem_H,axiom,(
    ! [V_a_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2)) = hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Orderings_Oord__class_Omin(T_b)),V_a_2),V_A_2) ) ) )).

fof(fact_Min__insert,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
         => c_Big__Operators_Olinorder__class_OMin(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),V_x_2),c_Big__Operators_Olinorder__class_OMin(T_b,V_A_2)) ) ) ) )).

fof(fact_Min_Osubset__idem,axiom,(
    ! [V_B_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( V_B_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
         => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2))
           => hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),c_Big__Operators_Olinorder__class_OMin(T_b,V_B_2)),c_Big__Operators_Olinorder__class_OMin(T_b,V_A_2)) = c_Big__Operators_Olinorder__class_OMin(T_b,V_A_2) ) ) ) ) )).

fof(fact_Min_Oeq__fold__idem_H,axiom,(
    ! [V_a_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => c_Big__Operators_Olinorder__class_OMin(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2)) = hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Orderings_Oord__class_Omin(T_b)),V_a_2),V_A_2) ) ) )).

fof(fact_card__insert,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
     => hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(c_Nat_OSuc,hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))) ) )).

fof(fact_min__max_OInf__fin_Oinsert,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( ~ hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
         => ( V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
           => c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),V_x_2),c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),V_A_2)) ) ) ) ) )).

fof(fact_min__max_OInf__fin_Oinsert__remove,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( ( hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
           => c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = V_x_2 )
          & ( hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
           => c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),V_x_2),c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))) ) ) ) ) )).

fof(fact_min__max_OInf__fin_Oeq__fold_H,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( ~ hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
         => c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Orderings_Oord__class_Omin(T_b)),V_x_2),V_A_2) ) ) ) )).

fof(fact_Min_Oinsert,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( ~ hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
         => ( V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
           => c_Big__Operators_Olinorder__class_OMin(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),V_x_2),c_Big__Operators_Olinorder__class_OMin(T_b,V_A_2)) ) ) ) ) )).

fof(fact_Min_Oinsert__remove,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( ( hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
           => c_Big__Operators_Olinorder__class_OMin(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = V_x_2 )
          & ( hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
           => c_Big__Operators_Olinorder__class_OMin(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),V_x_2),c_Big__Operators_Olinorder__class_OMin(T_b,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))) ) ) ) ) )).

fof(fact_Min_Oeq__fold_H,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( ~ hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
         => c_Big__Operators_Olinorder__class_OMin(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Orderings_Oord__class_Omin(T_b)),V_x_2),V_A_2) ) ) ) )).

fof(fact_diff__Suc__eq__diff__pred,axiom,(
    ! [V_n,V_m] : hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),hAPP(c_Nat_OSuc,V_n)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),c_Groups_Oone__class_Oone(tc_Nat_Onat))),V_n) )).

fof(fact_Suc__eq__plus1__left,axiom,(
    ! [V_n] : hAPP(c_Nat_OSuc,V_n) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),V_n) )).

fof(fact_Suc__eq__plus1,axiom,(
    ! [V_n] : hAPP(c_Nat_OSuc,V_n) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),c_Groups_Oone__class_Oone(tc_Nat_Onat)) )).

fof(fact_min__max_OInf__fin_Oclosed,axiom,(
    ! [V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
         => ( ! [B_x,B_y] : hBOOL(hAPP(hAPP(c_member(T_b),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),B_x),B_y)),hAPP(hAPP(c_Set_Oinsert(T_b),B_x),hAPP(hAPP(c_Set_Oinsert(T_b),B_y),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))))
           => hBOOL(hAPP(hAPP(c_member(T_b),c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),V_A_2)),V_A_2)) ) ) ) ) )).

fof(fact_Min_Oclosed,axiom,(
    ! [V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
         => ( ! [B_x,B_y] : hBOOL(hAPP(hAPP(c_member(T_b),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),B_x),B_y)),hAPP(hAPP(c_Set_Oinsert(T_b),B_x),hAPP(hAPP(c_Set_Oinsert(T_b),B_y),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))))
           => hBOOL(hAPP(hAPP(c_member(T_b),c_Big__Operators_Olinorder__class_OMin(T_b,V_A_2)),V_A_2)) ) ) ) ) )).

fof(fact_Max_Oremove,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
         => ( ( hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
             => c_Big__Operators_Olinorder__class_OMax(T_b,V_A_2) = V_x_2 )
            & ( hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
             => c_Big__Operators_Olinorder__class_OMax(T_b,V_A_2) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),V_x_2),c_Big__Operators_Olinorder__class_OMax(T_b,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))) ) ) ) ) ) )).

fof(fact_convex__bound__lt,axiom,(
    ! [V_v,V_u,V_y,V_a,V_x,T_a] :
      ( class_Rings_Olinordered__semiring__1__strict(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_a))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_a))
         => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_u))
           => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_v))
             => ( hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_u),V_v) = c_Groups_Oone__class_Oone(T_a)
               => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_u),V_x)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_v),V_y))),V_a)) ) ) ) ) ) ) )).

fof(fact_fold__image__1,axiom,(
    ! [V_f_2,V_S_2,T_b,T_c] :
      ( class_Groups_Ocomm__monoid__mult(T_c)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_S_2))
       => ( ! [B_x] :
              ( hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_S_2))
             => hAPP(V_f_2,B_x) = c_Groups_Oone__class_Oone(T_c) )
         => hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_c,T_b,c_Groups_Otimes__class_Otimes(T_c)),V_f_2),c_Groups_Oone__class_Oone(T_c)),V_S_2) = c_Groups_Oone__class_Oone(T_c) ) ) ) )).

fof(fact_less__zeroE,axiom,(
    ! [V_n] : ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) )).

fof(fact_le0,axiom,(
    ! [V_n] : hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n)) )).

fof(fact_zero__less__Suc,axiom,(
    ! [V_n] : hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Nat_OSuc,V_n))) )).

fof(fact_min__max_Osup__assoc,axiom,(
    ! [V_z,V_y,V_x,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),V_y)),V_z) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_y),V_z)) ) )).

fof(fact_min__max_Osup_Oassoc,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_a),V_b)),V_c) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_a),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_b),V_c)) ) )).

fof(fact_min__max_Osup__left__commute,axiom,(
    ! [V_z,V_y,V_x,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_y),V_z)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_y),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),V_z)) ) )).

fof(fact_min__max_Osup_Oleft__commute,axiom,(
    ! [V_c,V_a,V_b,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_b),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_a),V_c)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_a),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_b),V_c)) ) )).

fof(fact_min__max_Osup__left__idem,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),V_y)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),V_y) ) )).

fof(fact_min__max_Osup_Oleft__idem,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_a),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_a),V_b)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_a),V_b) ) )).

fof(fact_min__max_Osup__commute,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),V_y) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_y),V_x) ) )).

fof(fact_min__max_Osup_Ocommute,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_a),V_b) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_b),V_a) ) )).

fof(fact_min__max_Osup_Oidem,axiom,(
    ! [V_a,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_a),V_a) = V_a ) )).

fof(fact_zero__reorient,axiom,(
    ! [V_x_2,T_b] :
      ( class_Groups_Ozero(T_b)
     => ( c_Groups_Ozero__class_Ozero(T_b) = V_x_2
      <=> V_x_2 = c_Groups_Ozero__class_Ozero(T_b) ) ) )).

fof(fact_Max_Oidem,axiom,(
    ! [V_x,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),V_x) = V_x ) )).

fof(fact_le__max__iff__disj,axiom,(
    ! [V_y_2,V_x_2,V_z_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_z_2),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),V_x_2),V_y_2)))
      <=> ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_z_2),V_x_2))
          | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_z_2),V_y_2)) ) ) ) )).

fof(fact_le__maxI1,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),V_y))) ) )).

fof(fact_le__maxI2,axiom,(
    ! [V_x,V_y,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),V_y))) ) )).

fof(fact_min__max_Ole__iff__sup,axiom,(
    ! [V_y_2,V_x_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),V_y_2))
      <=> hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),V_x_2),V_y_2) = V_y_2 ) ) )).

fof(fact_min__max_Ole__sup__iff,axiom,(
    ! [V_z_2,V_y_2,V_x_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),V_x_2),V_y_2)),V_z_2))
      <=> ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),V_z_2))
          & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_y_2),V_z_2)) ) ) ) )).

fof(fact_min__max_Ole__supI1,axiom,(
    ! [V_b,V_a,V_x,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_a))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_a),V_b))) ) ) )).

fof(fact_min__max_Ole__supI2,axiom,(
    ! [V_a,V_b,V_x,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_b))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_a),V_b))) ) ) )).

fof(fact_min__max_Osup__absorb2,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y))
       => hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),V_y) = V_y ) ) )).

fof(fact_min__max_Osup__absorb1,axiom,(
    ! [V_x,V_y,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),V_x))
       => hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),V_y) = V_x ) ) )).

fof(fact_min__max_Ole__supI,axiom,(
    ! [V_b,V_x,V_a,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_x))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),V_x))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_a),V_b)),V_x)) ) ) ) )).

fof(fact_min__max_Osup__least,axiom,(
    ! [V_z,V_x,V_y,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),V_x))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_z),V_x))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_y),V_z)),V_x)) ) ) ) )).

fof(fact_min__max_Osup__mono,axiom,(
    ! [V_d,V_b,V_c,V_a,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_c))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),V_d))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_a),V_b)),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_c),V_d))) ) ) ) )).

fof(fact_min__max_Ole__supE,axiom,(
    ! [V_x,V_b,V_a,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_a),V_b)),V_x))
       => ~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_x))
           => ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),V_x)) ) ) ) )).

fof(fact_min__max_Oless__supI2,axiom,(
    ! [V_a,V_b,V_x,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_b))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_a),V_b))) ) ) )).

fof(fact_min__max_Oless__supI1,axiom,(
    ! [V_b,V_a,V_x,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_a))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_a),V_b))) ) ) )).

fof(fact_max__less__iff__conj,axiom,(
    ! [V_z_2,V_y_2,V_x_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),V_x_2),V_y_2)),V_z_2))
      <=> ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_x_2),V_z_2))
          & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_y_2),V_z_2)) ) ) ) )).

fof(fact_less__max__iff__disj,axiom,(
    ! [V_y_2,V_x_2,V_z_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_z_2),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),V_x_2),V_y_2)))
      <=> ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_z_2),V_x_2))
          | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_z_2),V_y_2)) ) ) ) )).

fof(fact_max__add__distrib__left,axiom,(
    ! [V_z,V_y,V_x,T_a] :
      ( class_Groups_Oordered__ab__semigroup__add__imp__le(T_a)
     => hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),V_y)),V_z) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_x),V_z)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_y),V_z)) ) )).

fof(fact_max__diff__distrib__left,axiom,(
    ! [V_z,V_y,V_x,T_a] :
      ( class_Groups_Oordered__ab__group__add(T_a)
     => hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),V_y)),V_z) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_x),V_z)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_y),V_z)) ) )).

fof(fact_max__Suc__Suc,axiom,(
    ! [V_n,V_m] : hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),hAPP(c_Nat_OSuc,V_n)) = hAPP(c_Nat_OSuc,hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),V_m),V_n)) )).

fof(fact_min__max_Oinf__sup__distrib2,axiom,(
    ! [V_x,V_z,V_y,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_y),V_z)),V_x) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_y),V_x)),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_z),V_x)) ) )).

fof(fact_min__max_Osup__inf__distrib2,axiom,(
    ! [V_x,V_z,V_y,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_y),V_z)),V_x) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_y),V_x)),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_z),V_x)) ) )).

fof(fact_min__max_Oinf__sup__distrib1,axiom,(
    ! [V_z,V_y,V_x,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_y),V_z)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),V_y)),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),V_z)) ) )).

fof(fact_min__max_Osup__inf__distrib1,axiom,(
    ! [V_z,V_y,V_x,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_y),V_z)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),V_y)),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),V_z)) ) )).

fof(fact_min__max_Oinf__sup__absorb,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),V_y)) = V_x ) )).

fof(fact_min__max_Osup__inf__absorb,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),V_y)) = V_x ) )).

fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,(
    ! [V_a,T_a] :
      ( class_Rings_Ocomm__semiring__1(T_a)
     => hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a) = V_a ) )).

fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,(
    ! [V_a,T_a] :
      ( class_Rings_Ocomm__semiring__1(T_a)
     => hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a)) = V_a ) )).

fof(fact_add__0__iff,axiom,(
    ! [V_a_2,V_b_2,T_b] :
      ( class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(T_b)
     => ( V_b_2 = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_b_2),V_a_2)
      <=> V_a_2 = c_Groups_Ozero__class_Ozero(T_b) ) ) )).

fof(fact_add_Ocomm__neutral,axiom,(
    ! [V_a,T_a] :
      ( class_Groups_Ocomm__monoid__add(T_a)
     => hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a)) = V_a ) )).

fof(fact_add__0__right,axiom,(
    ! [V_a,T_a] :
      ( class_Groups_Omonoid__add(T_a)
     => hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a)) = V_a ) )).

fof(fact_double__zero__sym,axiom,(
    ! [V_a_2,T_b] :
      ( class_Groups_Olinordered__ab__group__add(T_b)
     => ( c_Groups_Ozero__class_Ozero(T_b) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_a_2),V_a_2)
      <=> V_a_2 = c_Groups_Ozero__class_Ozero(T_b) ) ) )).

fof(fact_add__0,axiom,(
    ! [V_a,T_a] :
      ( class_Groups_Ocomm__monoid__add(T_a)
     => hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a) = V_a ) )).

fof(fact_add__0__left,axiom,(
    ! [V_a,T_a] :
      ( class_Groups_Omonoid__add(T_a)
     => hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a) = V_a ) )).

fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,(
    ! [V_a,T_a] :
      ( class_Rings_Ocomm__semiring__1(T_a)
     => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a) = c_Groups_Ozero__class_Ozero(T_a) ) )).

fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,(
    ! [V_a,T_a] :
      ( class_Rings_Ocomm__semiring__1(T_a)
     => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a)) = c_Groups_Ozero__class_Ozero(T_a) ) )).

fof(fact_mult__zero__left,axiom,(
    ! [V_a,T_a] :
      ( class_Rings_Omult__zero(T_a)
     => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a) = c_Groups_Ozero__class_Ozero(T_a) ) )).

fof(fact_mult__zero__right,axiom,(
    ! [V_a,T_a] :
      ( class_Rings_Omult__zero(T_a)
     => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a)) = c_Groups_Ozero__class_Ozero(T_a) ) )).

fof(fact_mult__eq__0__iff,axiom,(
    ! [V_b_2,V_a_2,T_b] :
      ( class_Rings_Oring__no__zero__divisors(T_b)
     => ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_b_2) = c_Groups_Ozero__class_Ozero(T_b)
      <=> ( V_a_2 = c_Groups_Ozero__class_Ozero(T_b)
          | V_b_2 = c_Groups_Ozero__class_Ozero(T_b) ) ) ) )).

fof(fact_no__zero__divisors,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Rings_Ono__zero__divisors(T_a)
     => ( V_a != c_Groups_Ozero__class_Ozero(T_a)
       => ( V_b != c_Groups_Ozero__class_Ozero(T_a)
         => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b) != c_Groups_Ozero__class_Ozero(T_a) ) ) ) )).

fof(fact_divisors__zero,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Rings_Ono__zero__divisors(T_a)
     => ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b) = c_Groups_Ozero__class_Ozero(T_a)
       => ( V_a = c_Groups_Ozero__class_Ozero(T_a)
          | V_b = c_Groups_Ozero__class_Ozero(T_a) ) ) ) )).

fof(fact_diff__0__right,axiom,(
    ! [V_a,T_a] :
      ( class_Groups_Ogroup__add(T_a)
     => hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a)) = V_a ) )).

fof(fact_diff__self,axiom,(
    ! [V_a,T_a] :
      ( class_Groups_Ogroup__add(T_a)
     => hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_a) = c_Groups_Ozero__class_Ozero(T_a) ) )).

fof(fact_eq__iff__diff__eq__0,axiom,(
    ! [V_b_2,V_a_2,T_b] :
      ( class_Groups_Oab__group__add(T_b)
     => ( V_a_2 = V_b_2
      <=> hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),V_a_2),V_b_2) = c_Groups_Ozero__class_Ozero(T_b) ) ) )).

fof(fact_right__minus__eq,axiom,(
    ! [V_b_2,V_a_2,T_b] :
      ( class_Groups_Ogroup__add(T_b)
     => ( hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),V_a_2),V_b_2) = c_Groups_Ozero__class_Ozero(T_b)
      <=> V_a_2 = V_b_2 ) ) )).

fof(fact_one__neq__zero,axiom,(
    ! [T_a] :
      ( class_Rings_Ozero__neq__one(T_a)
     => c_Groups_Oone__class_Oone(T_a) != c_Groups_Ozero__class_Ozero(T_a) ) )).

fof(fact_zero__neq__one,axiom,(
    ! [T_a] :
      ( class_Rings_Ozero__neq__one(T_a)
     => c_Groups_Ozero__class_Ozero(T_a) != c_Groups_Oone__class_Oone(T_a) ) )).

fof(fact_Suc__neq__Zero,axiom,(
    ! [V_m] : hAPP(c_Nat_OSuc,V_m) != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )).

fof(fact_Zero__neq__Suc,axiom,(
    ! [V_m] : c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(c_Nat_OSuc,V_m) )).

fof(fact_nat_Osimps_I3_J,axiom,(
    ! [V_nat_H_1] : hAPP(c_Nat_OSuc,V_nat_H_1) != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )).

fof(fact_Suc__not__Zero,axiom,(
    ! [V_m] : hAPP(c_Nat_OSuc,V_m) != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )).

fof(fact_nat_Osimps_I2_J,axiom,(
    ! [V_nat_H] : c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(c_Nat_OSuc,V_nat_H) )).

fof(fact_Zero__not__Suc,axiom,(
    ! [V_m] : c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(c_Nat_OSuc,V_m) )).

fof(fact_not__less0,axiom,(
    ! [V_n] : ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) )).

fof(fact_neq0__conv,axiom,(
    ! [V_n_2] :
      ( V_n_2 != c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n_2)) ) )).

fof(fact_less__nat__zero__code,axiom,(
    ! [V_n] : ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) )).

fof(fact_gr__implies__not0,axiom,(
    ! [V_n,V_m] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n))
     => V_n != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )).

fof(fact_gr0I,axiom,(
    ! [V_n] :
      ( V_n != c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n)) ) )).

fof(fact_plus__nat_Oadd__0,axiom,(
    ! [V_n] : hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n) = V_n )).

fof(fact_Nat_Oadd__0__right,axiom,(
    ! [V_m] : hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = V_m )).

fof(fact_add__is__0,axiom,(
    ! [V_n_2,V_m_2] :
      ( hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m_2),V_n_2) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
    <=> ( V_m_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
        & V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) ) )).

fof(fact_add__eq__self__zero,axiom,(
    ! [V_n,V_m] :
      ( hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n) = V_m
     => V_n = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )).

fof(fact_less__eq__nat_Osimps_I1_J,axiom,(
    ! [V_n] : hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n)) )).

fof(fact_le__0__eq,axiom,(
    ! [V_n_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n_2),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))
    <=> V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )).

fof(fact_diff__0__eq__0,axiom,(
    ! [V_n] : hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )).

fof(fact_minus__nat_Odiff__0,axiom,(
    ! [V_m] : hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = V_m )).

fof(fact_diff__self__eq__0,axiom,(
    ! [V_m] : hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_m) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )).

fof(fact_diffs0__imp__equal,axiom,(
    ! [V_n,V_m] :
      ( hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
     => ( hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),V_m) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
       => V_m = V_n ) ) )).

fof(fact_mult__0,axiom,(
    ! [V_n] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )).

fof(fact_mult__0__right,axiom,(
    ! [V_m] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )).

fof(fact_mult__is__0,axiom,(
    ! [V_n_2,V_m_2] :
      ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m_2),V_n_2) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
    <=> ( V_m_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
        | V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) ) )).

fof(fact_mult__cancel1,axiom,(
    ! [V_n_2,V_m_2,V_k_2] :
      ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_m_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_n_2)
    <=> ( V_m_2 = V_n_2
        | V_k_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) ) )).

fof(fact_mult__cancel2,axiom,(
    ! [V_n_2,V_k_2,V_m_2] :
      ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m_2),V_k_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n_2),V_k_2)
    <=> ( V_m_2 = V_n_2
        | V_k_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) ) )).

fof(fact_nat__mult__eq__cancel__disj,axiom,(
    ! [V_n_2,V_m_2,V_k_2] :
      ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_m_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_n_2)
    <=> ( V_k_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
        | V_m_2 = V_n_2 ) ) )).

fof(fact_sum__squares__eq__zero__iff,axiom,(
    ! [V_y_2,V_x_2,T_b] :
      ( class_Rings_Olinordered__ring__strict(T_b)
     => ( hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_x_2),V_x_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_y_2),V_y_2)) = c_Groups_Ozero__class_Ozero(T_b)
      <=> ( V_x_2 = c_Groups_Ozero__class_Ozero(T_b)
          & V_y_2 = c_Groups_Ozero__class_Ozero(T_b) ) ) ) )).

fof(fact_min__max_Ofun__left__comm__idem__sup,axiom,(
    ! [T_b] :
      ( class_Orderings_Olinorder(T_b)
     => c_Finite__Set_Ofun__left__comm__idem(T_b,T_b,c_Orderings_Oord__class_Omax(T_b)) ) )).

fof(fact_min__0L,axiom,(
    ! [V_n] : hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )).

fof(fact_min__0R,axiom,(
    ! [V_n] : hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),V_n),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )).

fof(fact_min__max_Odistrib__inf__le,axiom,(
    ! [V_z,V_y,V_x,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),V_y)),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),V_z))),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_y),V_z)))) ) )).

fof(fact_min__max_Odistrib__sup__le,axiom,(
    ! [V_z,V_y,V_x,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_y),V_z))),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),V_y)),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),V_z)))) ) )).

fof(fact_nat__minus__add__max,axiom,(
    ! [V_m,V_n] : hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),V_m)),V_m) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),V_n),V_m) )).

fof(fact_sum__squares__ge__zero,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Rings_Olinordered__ring(T_a)
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_x)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_y),V_y)))) ) )).

fof(fact_sum__squares__le__zero__iff,axiom,(
    ! [V_y_2,V_x_2,T_b] :
      ( class_Rings_Olinordered__ring__strict(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_x_2),V_x_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_y_2),V_y_2))),c_Groups_Ozero__class_Ozero(T_b)))
      <=> ( V_x_2 = c_Groups_Ozero__class_Ozero(T_b)
          & V_y_2 = c_Groups_Ozero__class_Ozero(T_b) ) ) ) )).

fof(fact_sum__squares__gt__zero__iff,axiom,(
    ! [V_y_2,V_x_2,T_b] :
      ( class_Rings_Olinordered__ring__strict(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_x_2),V_x_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_y_2),V_y_2))))
      <=> ( V_x_2 != c_Groups_Ozero__class_Ozero(T_b)
          | V_y_2 != c_Groups_Ozero__class_Ozero(T_b) ) ) ) )).

fof(fact_not__sum__squares__lt__zero,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Rings_Olinordered__ring(T_a)
     => ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_x)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_y),V_y))),c_Groups_Ozero__class_Ozero(T_a))) ) )).

fof(fact_zero__le__double__add__iff__zero__le__single__add,axiom,(
    ! [V_a_2,T_b] :
      ( class_Groups_Olinordered__ab__group__add(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_a_2),V_a_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_a_2)) ) ) )).

fof(fact_double__add__le__zero__iff__single__add__le__zero,axiom,(
    ! [V_a_2,T_b] :
      ( class_Groups_Olinordered__ab__group__add(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_a_2),V_a_2)),c_Groups_Ozero__class_Ozero(T_b)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),c_Groups_Ozero__class_Ozero(T_b))) ) ) )).

fof(fact_add__nonneg__nonneg,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Groups_Oordered__comm__monoid__add(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_b))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b))) ) ) ) )).

fof(fact_add__nonneg__eq__0__iff,axiom,(
    ! [V_y_2,V_x_2,T_b] :
      ( class_Groups_Oordered__comm__monoid__add(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_x_2))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_y_2))
         => ( hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_x_2),V_y_2) = c_Groups_Ozero__class_Ozero(T_b)
          <=> ( V_x_2 = c_Groups_Ozero__class_Ozero(T_b)
              & V_y_2 = c_Groups_Ozero__class_Ozero(T_b) ) ) ) ) ) )).

fof(fact_add__increasing,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Groups_Oordered__comm__monoid__add(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),V_c))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c))) ) ) ) )).

fof(fact_add__increasing2,axiom,(
    ! [V_a,V_b,V_c,T_a] :
      ( class_Groups_Oordered__comm__monoid__add(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),V_a))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c))) ) ) ) )).

fof(fact_add__nonpos__nonpos,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Groups_Oordered__comm__monoid__add(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a)))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),c_Groups_Ozero__class_Ozero(T_a)))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),c_Groups_Ozero__class_Ozero(T_a))) ) ) ) )).

fof(fact_zero__le__square,axiom,(
    ! [V_a,T_a] :
      ( class_Rings_Olinordered__ring(T_a)
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_a))) ) )).

fof(fact_zero__le__mult__iff,axiom,(
    ! [V_b_2,V_a_2,T_b] :
      ( class_Rings_Olinordered__ring__strict(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_b_2)))
      <=> ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_a_2))
            & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_b_2)) )
          | ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),c_Groups_Ozero__class_Ozero(T_b)))
            & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_b_2),c_Groups_Ozero__class_Ozero(T_b))) ) ) ) ) )).

fof(fact_mult__le__0__iff,axiom,(
    ! [V_b_2,V_a_2,T_b] :
      ( class_Rings_Olinordered__ring__strict(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_b_2)),c_Groups_Ozero__class_Ozero(T_b)))
      <=> ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_a_2))
            & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_b_2),c_Groups_Ozero__class_Ozero(T_b))) )
          | ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),c_Groups_Ozero__class_Ozero(T_b)))
            & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_b_2)) ) ) ) ) )).

fof(fact_mult__nonneg__nonneg,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Rings_Oordered__cancel__semiring(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_b))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b))) ) ) ) )).

fof(fact_mult__nonneg__nonpos,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Rings_Oordered__cancel__semiring(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),c_Groups_Ozero__class_Ozero(T_a)))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)),c_Groups_Ozero__class_Ozero(T_a))) ) ) ) )).

fof(fact_mult__nonneg__nonpos2,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Rings_Oordered__cancel__semiring(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),c_Groups_Ozero__class_Ozero(T_a)))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a)),c_Groups_Ozero__class_Ozero(T_a))) ) ) ) )).

fof(fact_mult__nonpos__nonneg,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Rings_Oordered__cancel__semiring(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a)))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_b))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)),c_Groups_Ozero__class_Ozero(T_a))) ) ) ) )).

fof(fact_mult__nonpos__nonpos,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Rings_Oordered__ring(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a)))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),c_Groups_Ozero__class_Ozero(T_a)))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b))) ) ) ) )).

fof(fact_mult__right__mono,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Rings_Oordered__semiring(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c))) ) ) ) )).

fof(fact_mult__left__mono,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Rings_Oordered__semiring(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b))) ) ) ) )).

fof(fact_comm__mult__left__mono,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Rings_Oordered__comm__semiring(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b))) ) ) ) )).

fof(fact_mult__right__mono__neg,axiom,(
    ! [V_c,V_a,V_b,T_a] :
      ( class_Rings_Oordered__ring(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),V_a))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_c),c_Groups_Ozero__class_Ozero(T_a)))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c))) ) ) ) )).

fof(fact_mult__left__mono__neg,axiom,(
    ! [V_c,V_a,V_b,T_a] :
      ( class_Rings_Oordered__ring(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),V_a))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_c),c_Groups_Ozero__class_Ozero(T_a)))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b))) ) ) ) )).

fof(fact_mult__mono_H,axiom,(
    ! [V_d,V_c,V_b,V_a,T_a] :
      ( class_Rings_Oordered__semiring(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_c),V_d))
         => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a))
           => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c))
             => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_d))) ) ) ) ) ) )).

fof(fact_mult__mono,axiom,(
    ! [V_d,V_c,V_b,V_a,T_a] :
      ( class_Rings_Oordered__semiring(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_c),V_d))
         => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_b))
           => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c))
             => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_d))) ) ) ) ) ) )).

fof(fact_split__mult__pos__le,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Rings_Oordered__ring(T_a)
     => ( ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a))
            & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_b)) )
          | ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a)))
            & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),c_Groups_Ozero__class_Ozero(T_a))) ) )
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b))) ) ) )).

fof(fact_split__mult__neg__le,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Rings_Oordered__cancel__semiring(T_a)
     => ( ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a))
            & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),c_Groups_Ozero__class_Ozero(T_a))) )
          | ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a)))
            & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_b)) ) )
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)),c_Groups_Ozero__class_Ozero(T_a))) ) ) )).

fof(fact_add__neg__neg,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Groups_Oordered__comm__monoid__add(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a)))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),c_Groups_Ozero__class_Ozero(T_a)))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),c_Groups_Ozero__class_Ozero(T_a))) ) ) ) )).

fof(fact_add__pos__pos,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Groups_Oordered__comm__monoid__add(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_b))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b))) ) ) ) )).

fof(fact_double__add__less__zero__iff__single__add__less__zero,axiom,(
    ! [V_a_2,T_b] :
      ( class_Groups_Olinordered__ab__group__add(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_a_2),V_a_2)),c_Groups_Ozero__class_Ozero(T_b)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),c_Groups_Ozero__class_Ozero(T_b))) ) ) )).

fof(fact_zero__less__double__add__iff__zero__less__single__add,axiom,(
    ! [V_a_2,T_b] :
      ( class_Groups_Olinordered__ab__group__add(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_a_2),V_a_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_a_2)) ) ) )).

fof(fact_pos__add__strict,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Rings_Olinordered__semidom(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),V_c))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c))) ) ) ) )).

fof(fact_mult__strict__left__mono__neg,axiom,(
    ! [V_c,V_a,V_b,T_a] :
      ( class_Rings_Olinordered__ring__strict(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),V_a))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_c),c_Groups_Ozero__class_Ozero(T_a)))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b))) ) ) ) )).

fof(fact_mult__strict__right__mono__neg,axiom,(
    ! [V_c,V_a,V_b,T_a] :
      ( class_Rings_Olinordered__ring__strict(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),V_a))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_c),c_Groups_Ozero__class_Ozero(T_a)))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c))) ) ) ) )).

fof(fact_comm__mult__strict__left__mono,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Rings_Olinordered__comm__semiring__strict(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b))) ) ) ) )).

fof(fact_mult__strict__left__mono,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Rings_Olinordered__semiring__strict(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b))) ) ) ) )).

fof(fact_mult__strict__right__mono,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Rings_Olinordered__semiring__strict(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c))) ) ) ) )).

fof(fact_mult__neg__neg,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Rings_Olinordered__ring__strict(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a)))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),c_Groups_Ozero__class_Ozero(T_a)))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b))) ) ) ) )).

fof(fact_mult__neg__pos,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Rings_Olinordered__semiring__strict(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a)))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_b))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)),c_Groups_Ozero__class_Ozero(T_a))) ) ) ) )).

fof(fact_mult__less__cancel__left__neg,axiom,(
    ! [V_b_2,V_a_2,V_c_2,T_b] :
      ( class_Rings_Olinordered__ring__strict(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),c_Groups_Ozero__class_Ozero(T_b)))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_c_2),V_a_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_c_2),V_b_2)))
        <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_b_2),V_a_2)) ) ) ) )).

fof(fact_zero__less__mult__pos2,axiom,(
    ! [V_a,V_b,T_a] :
      ( class_Rings_Olinordered__semiring__strict(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a)))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_b)) ) ) ) )).

fof(fact_zero__less__mult__pos,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Rings_Olinordered__semiring__strict(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_b)) ) ) ) )).

fof(fact_mult__pos__neg2,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Rings_Olinordered__semiring__strict(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),c_Groups_Ozero__class_Ozero(T_a)))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a)),c_Groups_Ozero__class_Ozero(T_a))) ) ) ) )).

fof(fact_mult__pos__neg,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Rings_Olinordered__semiring__strict(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),c_Groups_Ozero__class_Ozero(T_a)))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)),c_Groups_Ozero__class_Ozero(T_a))) ) ) ) )).

fof(fact_mult__pos__pos,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Rings_Olinordered__semiring__strict(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_b))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b))) ) ) ) )).

fof(fact_mult__less__cancel__left__pos,axiom,(
    ! [V_b_2,V_a_2,V_c_2,T_b] :
      ( class_Rings_Olinordered__ring__strict(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_c_2))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_c_2),V_a_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_c_2),V_b_2)))
        <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),V_b_2)) ) ) ) )).

fof(fact_mult__less__cancel__left__disj,axiom,(
    ! [V_b_2,V_a_2,V_c_2,T_b] :
      ( class_Rings_Olinordered__ring__strict(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_c_2),V_a_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_c_2),V_b_2)))
      <=> ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_c_2))
            & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),V_b_2)) )
          | ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),c_Groups_Ozero__class_Ozero(T_b)))
            & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_b_2),V_a_2)) ) ) ) ) )).

fof(fact_mult__less__cancel__right__disj,axiom,(
    ! [V_b_2,V_c_2,V_a_2,T_b] :
      ( class_Rings_Olinordered__ring__strict(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_c_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_b_2),V_c_2)))
      <=> ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_c_2))
            & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),V_b_2)) )
          | ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),c_Groups_Ozero__class_Ozero(T_b)))
            & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_b_2),V_a_2)) ) ) ) ) )).

fof(fact_not__square__less__zero,axiom,(
    ! [V_a,T_a] :
      ( class_Rings_Olinordered__ring(T_a)
     => ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_a)),c_Groups_Ozero__class_Ozero(T_a))) ) )).

fof(fact_le__iff__diff__le__0,axiom,(
    ! [V_b_2,V_a_2,T_b] :
      ( class_Groups_Oordered__ab__group__add(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),V_b_2))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),V_a_2),V_b_2)),c_Groups_Ozero__class_Ozero(T_b))) ) ) )).

fof(fact_less__iff__diff__less__0,axiom,(
    ! [V_b_2,V_a_2,T_b] :
      ( class_Groups_Oordered__ab__group__add(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),V_b_2))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),V_a_2),V_b_2)),c_Groups_Ozero__class_Ozero(T_b))) ) ) )).

fof(fact_add__scale__eq__noteq,axiom,(
    ! [V_d,V_c,V_b,V_a,V_r,T_a] :
      ( class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(T_a)
     => ( V_r != c_Groups_Ozero__class_Ozero(T_a)
       => ( ( V_a = V_b
            & V_c != V_d )
         => hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_r),V_c)) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_r),V_d)) ) ) ) )).

fof(fact_not__one__le__zero,axiom,(
    ! [T_a] :
      ( class_Rings_Olinordered__semidom(T_a)
     => ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Oone__class_Oone(T_a)),c_Groups_Ozero__class_Ozero(T_a))) ) )).

fof(fact_zero__le__one,axiom,(
    ! [T_a] :
      ( class_Rings_Olinordered__semidom(T_a)
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),c_Groups_Oone__class_Oone(T_a))) ) )).

fof(fact_zero__less__one,axiom,(
    ! [T_a] :
      ( class_Rings_Olinordered__semidom(T_a)
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),c_Groups_Oone__class_Oone(T_a))) ) )).

fof(fact_not__one__less__zero,axiom,(
    ! [T_a] :
      ( class_Rings_Olinordered__semidom(T_a)
     => ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Oone__class_Oone(T_a)),c_Groups_Ozero__class_Ozero(T_a))) ) )).

fof(fact_setsum_Oempty,axiom,(
    ! [V_g_2,T_c,T_b] :
      ( class_Groups_Ocomm__monoid__add(T_b)
     => hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_b),V_g_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) = c_Groups_Ozero__class_Ozero(T_b) ) )).

fof(fact_setsum__empty,axiom,(
    ! [V_f_2,T_c,T_b] :
      ( class_Groups_Ocomm__monoid__add(T_b)
     => hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_b),V_f_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) = c_Groups_Ozero__class_Ozero(T_b) ) )).

fof(fact_setsum_Oinfinite,axiom,(
    ! [V_g_2,V_A_2,T_b,T_c] :
      ( class_Groups_Ocomm__monoid__add(T_c)
     => ( ~ hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_g_2),V_A_2) = c_Groups_Ozero__class_Ozero(T_c) ) ) )).

fof(fact_setsum__infinite,axiom,(
    ! [V_f_2,V_A_2,T_b,T_c] :
      ( class_Groups_Ocomm__monoid__add(T_c)
     => ( ~ hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),V_A_2) = c_Groups_Ozero__class_Ozero(T_c) ) ) )).

fof(fact_setprod__zero,axiom,(
    ! [V_f_2,V_A_2,T_b,T_c] :
      ( class_Rings_Ocomm__semiring__1(T_c)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( ? [B_x] :
              ( hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2))
              & hAPP(V_f_2,B_x) = c_Groups_Ozero__class_Ozero(T_c) )
         => hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_f_2),V_A_2) = c_Groups_Ozero__class_Ozero(T_c) ) ) ) )).

fof(fact_setprod__zero__iff,axiom,(
    ! [V_f_2,V_A_2,T_b,T_c] :
      ( ( class_Rings_Ocomm__semiring__1(T_c)
        & class_Rings_Ono__zero__divisors(T_c) )
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_f_2),V_A_2) = c_Groups_Ozero__class_Ozero(T_c)
        <=> ? [B_x] :
              ( hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2))
              & hAPP(V_f_2,B_x) = c_Groups_Ozero__class_Ozero(T_c) ) ) ) ) )).

fof(fact_gr0__conv__Suc,axiom,(
    ! [V_n_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n_2))
    <=> ? [B_m] : V_n_2 = hAPP(c_Nat_OSuc,B_m) ) )).

fof(fact_less__Suc0,axiom,(
    ! [V_n_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))))
    <=> V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )).

fof(fact_less__Suc__eq__0__disj,axiom,(
    ! [V_n_2,V_m_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),hAPP(c_Nat_OSuc,V_n_2)))
    <=> ( V_m_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
        | ? [B_j] :
            ( V_m_2 = hAPP(c_Nat_OSuc,B_j)
            & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_j),V_n_2)) ) ) ) )).

fof(fact_add__is__1,axiom,(
    ! [V_n_2,V_m_2] :
      ( hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m_2),V_n_2) = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))
    <=> ( ( V_m_2 = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))
          & V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )
        | ( V_m_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
          & V_n_2 = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) ) ) ) )).

fof(fact_one__is__add,axiom,(
    ! [V_n_2,V_m_2] :
      ( hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m_2),V_n_2)
    <=> ( ( V_m_2 = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))
          & V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )
        | ( V_m_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
          & V_n_2 = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) ) ) ) )).

fof(fact_add__gr__0,axiom,(
    ! [V_n_2,V_m_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m_2),V_n_2)))
    <=> ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_m_2))
        | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n_2)) ) ) )).

fof(fact_card_Oempty,axiom,(
    ! [T_b] : hAPP(c_Finite__Set_Ocard(T_b),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )).

fof(fact_card__infinite,axiom,(
    ! [V_A_2,T_b] :
      ( ~ hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
     => hAPP(c_Finite__Set_Ocard(T_b),V_A_2) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )).

fof(fact_mult__eq__1__iff,axiom,(
    ! [V_n_2,V_m_2] :
      ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m_2),V_n_2) = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))
    <=> ( V_m_2 = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))
        & V_n_2 = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) ) ) )).

fof(fact_zero__less__diff,axiom,(
    ! [V_m_2,V_n_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n_2),V_m_2)))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) ) )).

fof(fact_diff__less,axiom,(
    ! [V_m,V_n] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_m))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)),V_m)) ) ) )).

fof(fact_nat__mult__less__cancel1,axiom,(
    ! [V_n_2,V_m_2,V_k_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_k_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_m_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_n_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) ) ) )).

fof(fact_nat__mult__eq__cancel1,axiom,(
    ! [V_n_2,V_m_2,V_k_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_k_2))
     => ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_m_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_n_2)
      <=> V_m_2 = V_n_2 ) ) )).

fof(fact_mult__less__mono2,axiom,(
    ! [V_k,V_j,V_i] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),V_j))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_k))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_i)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_j))) ) ) )).

fof(fact_mult__less__mono1,axiom,(
    ! [V_k,V_j,V_i] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),V_j))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_k))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i),V_k)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j),V_k))) ) ) )).

fof(fact_mult__less__cancel2,axiom,(
    ! [V_n_2,V_k_2,V_m_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m_2),V_k_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n_2),V_k_2)))
    <=> ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_k_2))
        & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) ) ) )).

fof(fact_mult__less__cancel1,axiom,(
    ! [V_n_2,V_m_2,V_k_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_m_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_n_2)))
    <=> ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_k_2))
        & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) ) ) )).

fof(fact_nat__0__less__mult__iff,axiom,(
    ! [V_n_2,V_m_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m_2),V_n_2)))
    <=> ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_m_2))
        & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n_2)) ) ) )).

fof(fact_diff__add__0,axiom,(
    ! [V_m,V_n] : hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),V_m)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )).

fof(fact_diff__is__0__eq_H,axiom,(
    ! [V_n,V_m] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),V_n))
     => hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )).

fof(fact_diff__is__0__eq,axiom,(
    ! [V_n_2,V_m_2] :
      ( hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m_2),V_n_2) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2)) ) )).

fof(fact_One__nat__def,axiom,(
    c_Groups_Oone__class_Oone(tc_Nat_Onat) = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) )).

fof(fact_mult__eq__self__implies__10,axiom,(
    ! [V_n,V_m] :
      ( V_m = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n)
     => ( V_n = c_Groups_Oone__class_Oone(tc_Nat_Onat)
        | V_m = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) ) )).

fof(fact_setsum__eq__0__iff,axiom,(
    ! [V_f_2,V_F_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_F_2))
     => ( hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,tc_Nat_Onat),V_f_2),V_F_2) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
      <=> ! [B_x] :
            ( hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_F_2))
           => hAPP(V_f_2,B_x) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) ) ) )).

fof(fact_min__max_Ofold__sup__insert,axiom,(
    ! [V_a_2,V_b_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Orderings_Oord__class_Omax(T_b)),V_b_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),V_a_2),hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Orderings_Oord__class_Omax(T_b)),V_b_2),V_A_2)) ) ) )).

fof(fact_Max_Oin__idem,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
         => hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),V_x_2),c_Big__Operators_Olinorder__class_OMax(T_b,V_A_2)) = c_Big__Operators_Olinorder__class_OMax(T_b,V_A_2) ) ) ) )).

fof(fact_Suc__pred_H,axiom,(
    ! [V_n] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n))
     => V_n = hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),c_Groups_Oone__class_Oone(tc_Nat_Onat))) ) )).

fof(fact_add__eq__if,axiom,(
    ! [V_n,V_m] :
      ( ( V_m = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
       => hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n) = V_n )
      & ( V_m != c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
       => hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n) = hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),c_Groups_Oone__class_Oone(tc_Nat_Onat))),V_n)) ) ) )).

fof(fact_add__nonpos__neg,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Groups_Oordered__comm__monoid__add(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a)))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),c_Groups_Ozero__class_Ozero(T_a)))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),c_Groups_Ozero__class_Ozero(T_a))) ) ) ) )).

fof(fact_add__neg__nonpos,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Groups_Oordered__comm__monoid__add(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a)))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),c_Groups_Ozero__class_Ozero(T_a)))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),c_Groups_Ozero__class_Ozero(T_a))) ) ) ) )).

fof(fact_add__strict__increasing2,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Groups_Oordered__comm__monoid__add(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),V_c))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c))) ) ) ) )).

fof(fact_add__strict__increasing,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Groups_Oordered__comm__monoid__add(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),V_c))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c))) ) ) ) )).

fof(fact_add__nonneg__pos,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Groups_Oordered__comm__monoid__add(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_b))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b))) ) ) ) )).

fof(fact_add__pos__nonneg,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Groups_Oordered__comm__monoid__add(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_b))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b))) ) ) ) )).

fof(fact_mult__le__cancel__left__pos,axiom,(
    ! [V_b_2,V_a_2,V_c_2,T_b] :
      ( class_Rings_Olinordered__ring__strict(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_c_2))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_c_2),V_a_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_c_2),V_b_2)))
        <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),V_b_2)) ) ) ) )).

fof(fact_mult__le__cancel__left__neg,axiom,(
    ! [V_b_2,V_a_2,V_c_2,T_b] :
      ( class_Rings_Olinordered__ring__strict(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),c_Groups_Ozero__class_Ozero(T_b)))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_c_2),V_a_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_c_2),V_b_2)))
        <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_b_2),V_a_2)) ) ) ) )).

fof(fact_mult__strict__mono,axiom,(
    ! [V_d,V_c,V_b,V_a,T_a] :
      ( class_Rings_Olinordered__semiring__strict(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_c),V_d))
         => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_b))
           => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c))
             => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_d))) ) ) ) ) ) )).

fof(fact_mult__strict__mono_H,axiom,(
    ! [V_d,V_c,V_b,V_a,T_a] :
      ( class_Rings_Olinordered__semiring__strict(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_c),V_d))
         => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a))
           => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c))
             => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_d))) ) ) ) ) ) )).

fof(fact_mult__less__le__imp__less,axiom,(
    ! [V_d,V_c,V_b,V_a,T_a] :
      ( class_Rings_Olinordered__semiring__strict(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_c),V_d))
         => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a))
           => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c))
             => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_d))) ) ) ) ) ) )).

fof(fact_mult__le__less__imp__less,axiom,(
    ! [V_d,V_c,V_b,V_a,T_a] :
      ( class_Rings_Olinordered__semiring__strict(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_c),V_d))
         => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a))
           => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c))
             => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_d))) ) ) ) ) ) )).

fof(fact_mult__right__less__imp__less,axiom,(
    ! [V_b,V_c,V_a,T_a] :
      ( class_Rings_Olinordered__semiring(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c)))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)) ) ) ) )).

fof(fact_mult__less__imp__less__right,axiom,(
    ! [V_b,V_c,V_a,T_a] :
      ( class_Rings_Olinordered__semiring__strict(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c)))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)) ) ) ) )).

fof(fact_mult__left__less__imp__less,axiom,(
    ! [V_b,V_a,V_c,T_a] :
      ( class_Rings_Olinordered__semiring(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b)))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)) ) ) ) )).

fof(fact_mult__less__imp__less__left,axiom,(
    ! [V_b,V_a,V_c,T_a] :
      ( class_Rings_Olinordered__semiring__strict(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b)))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)) ) ) ) )).

fof(fact_mult__right__le__imp__le,axiom,(
    ! [V_b,V_c,V_a,T_a] :
      ( class_Rings_Olinordered__semiring__strict(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c)))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b)) ) ) ) )).

fof(fact_mult__left__le__imp__le,axiom,(
    ! [V_b,V_a,V_c,T_a] :
      ( class_Rings_Olinordered__semiring__strict(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b)))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b)) ) ) ) )).

fof(fact_mult__eq__if,axiom,(
    ! [V_n,V_m] :
      ( ( V_m = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
       => hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )
      & ( V_m != c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
       => hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),c_Groups_Oone__class_Oone(tc_Nat_Onat))),V_n)) ) ) )).

fof(fact_mult__right__le__one__le,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Rings_Olinordered__idom(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_x))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_y))
         => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),c_Groups_Oone__class_Oone(T_a)))
           => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_y)),V_x)) ) ) ) ) )).

fof(fact_mult__left__le__one__le,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Rings_Olinordered__idom(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_x))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_y))
         => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),c_Groups_Oone__class_Oone(T_a)))
           => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_y),V_x)),V_x)) ) ) ) ) )).

fof(fact_zero__less__two,axiom,(
    ! [T_a] :
      ( class_Rings_Olinordered__semidom(T_a)
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Groups_Oone__class_Oone(T_a)),c_Groups_Oone__class_Oone(T_a)))) ) )).

fof(fact_Max_OF__eq,axiom,(
    ! [V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => c_Big__Operators_Olinorder__class_OMax(T_b,V_A_2) = hAPP(c_Finite__Set_Ofold1(T_b,c_Orderings_Oord__class_Omax(T_b)),V_A_2) ) ) )).

fof(fact_card__eq__0__iff,axiom,(
    ! [V_A_2,T_b] :
      ( hAPP(c_Finite__Set_Ocard(T_b),V_A_2) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
    <=> ( V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
        | ~ hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) ) ) )).

fof(fact_card__ge__0__finite,axiom,(
    ! [V_A_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Finite__Set_Ocard(T_b),V_A_2)))
     => hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) ) )).

fof(fact_Suc__pred,axiom,(
    ! [V_n] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n))
     => hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) = V_n ) )).

fof(fact_diff__Suc__less,axiom,(
    ! [V_i,V_n] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),hAPP(c_Nat_OSuc,V_i))),V_n)) ) )).

fof(fact_n__less__m__mult__n,axiom,(
    ! [V_m,V_n] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),V_n))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),V_m))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n))) ) ) )).

fof(fact_n__less__n__mult__m,axiom,(
    ! [V_m,V_n] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),V_n))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),V_m))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n),V_m))) ) ) )).

fof(fact_one__less__mult,axiom,(
    ! [V_m,V_n] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),V_n))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),V_m))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n))) ) ) )).

fof(fact_nat__diff__split__asm,axiom,(
    ! [V_b_2,V_a_2,V_P_2] :
      ( hBOOL(hAPP(V_P_2,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_a_2),V_b_2)))
    <=> ~ ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_a_2),V_b_2))
            & ~ hBOOL(hAPP(V_P_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) )
          | ? [B_d] :
              ( V_a_2 = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_b_2),B_d)
              & ~ hBOOL(hAPP(V_P_2,B_d)) ) ) ) )).

fof(fact_nat__diff__split,axiom,(
    ! [V_b_2,V_a_2,V_P_2] :
      ( hBOOL(hAPP(V_P_2,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_a_2),V_b_2)))
    <=> ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_a_2),V_b_2))
         => hBOOL(hAPP(V_P_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) )
        & ! [B_d] :
            ( V_a_2 = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_b_2),B_d)
           => hBOOL(hAPP(V_P_2,B_d)) ) ) ) )).

fof(fact_one__le__mult__iff,axiom,(
    ! [V_n_2,V_m_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m_2),V_n_2)))
    <=> ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),V_m_2))
        & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),V_n_2)) ) ) )).

fof(fact_nat__mult__le__cancel1,axiom,(
    ! [V_n_2,V_m_2,V_k_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_k_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_m_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_n_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2)) ) ) )).

fof(fact_mult__le__cancel2,axiom,(
    ! [V_n_2,V_k_2,V_m_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m_2),V_k_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n_2),V_k_2)))
    <=> ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_k_2))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2)) ) ) )).

fof(fact_mult__le__cancel1,axiom,(
    ! [V_n_2,V_m_2,V_k_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_m_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_n_2)))
    <=> ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_k_2))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2)) ) ) )).

fof(fact_setsum__eq__Suc0__iff,axiom,(
    ! [V_f_2,V_A_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
     => ( hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,tc_Nat_Onat),V_f_2),V_A_2) = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))
      <=> ? [B_x] :
            ( hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2))
            & hAPP(V_f_2,B_x) = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))
            & ! [B_xa] :
                ( hBOOL(hAPP(hAPP(c_member(T_b),B_xa),V_A_2))
               => ( B_x != B_xa
                 => hAPP(V_f_2,B_xa) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) ) ) ) ) )).

fof(fact_setsum__eq__1__iff,axiom,(
    ! [V_f_2,V_A_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
     => ( hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,tc_Nat_Onat),V_f_2),V_A_2) = c_Groups_Oone__class_Oone(tc_Nat_Onat)
      <=> ? [B_x] :
            ( hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2))
            & hAPP(V_f_2,B_x) = c_Groups_Oone__class_Oone(tc_Nat_Onat)
            & ! [B_xa] :
                ( hBOOL(hAPP(hAPP(c_member(T_b),B_xa),V_A_2))
               => ( B_x != B_xa
                 => hAPP(V_f_2,B_xa) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) ) ) ) ) )).

fof(fact_setprod__pos__nat__iff,axiom,(
    ! [V_f_2,V_S_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_S_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,tc_Nat_Onat),V_f_2),V_S_2)))
      <=> ! [B_x] :
            ( hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_S_2))
           => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(V_f_2,B_x))) ) ) ) )).

fof(fact_min__max_Osup__le__fold__sup,axiom,(
    ! [V_b_2,V_a_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),V_a_2),V_b_2)),hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Orderings_Oord__class_Omax(T_b)),V_b_2),V_A_2))) ) ) ) )).

fof(fact_Max__insert,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
         => c_Big__Operators_Olinorder__class_OMax(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),V_x_2),c_Big__Operators_Olinorder__class_OMax(T_b,V_A_2)) ) ) ) )).

fof(fact_Max_Osubset__idem,axiom,(
    ! [V_B_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( V_B_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
         => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2))
           => hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),c_Big__Operators_Olinorder__class_OMax(T_b,V_B_2)),c_Big__Operators_Olinorder__class_OMax(T_b,V_A_2)) = c_Big__Operators_Olinorder__class_OMax(T_b,V_A_2) ) ) ) ) )).

fof(fact_min__max_Osup__Inf__absorb,axiom,(
    ! [V_a_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2))
         => hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),V_a_2),c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),V_A_2)) = V_a_2 ) ) ) )).

fof(fact_Max_Oeq__fold__idem_H,axiom,(
    ! [V_a_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => c_Big__Operators_Olinorder__class_OMax(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2)) = hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Orderings_Oord__class_Omax(T_b)),V_a_2),V_A_2) ) ) )).

fof(fact_convex__bound__le,axiom,(
    ! [V_v,V_u,V_y,V_a,V_x,T_a] :
      ( class_Rings_Olinordered__semiring__1(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_a))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),V_a))
         => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_u))
           => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_v))
             => ( hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_u),V_v) = c_Groups_Oone__class_Oone(T_a)
               => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_u),V_x)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_v),V_y))),V_a)) ) ) ) ) ) ) )).

fof(fact_card__gt__0__iff,axiom,(
    ! [V_A_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Finite__Set_Ocard(T_b),V_A_2)))
    <=> ( V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
        & hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) ) ) )).

fof(fact_setsum_Oeq__fold,axiom,(
    ! [V_g_2,V_A_2,T_b,T_c] :
      ( class_Groups_Ocomm__monoid__add(T_c)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_g_2),V_A_2) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_c,T_b,c_Groups_Oplus__class_Oplus(T_c)),V_g_2),c_Groups_Ozero__class_Ozero(T_c)),V_A_2) ) ) )).

fof(fact_setsum_OF__eq,axiom,(
    ! [V_g_2,V_A_2,T_b,T_c] :
      ( class_Groups_Ocomm__monoid__add(T_c)
     => ( ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
         => hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_g_2),V_A_2) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_c,T_b,c_Groups_Oplus__class_Oplus(T_c)),V_g_2),c_Groups_Ozero__class_Ozero(T_c)),V_A_2) )
        & ( ~ hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
         => hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_g_2),V_A_2) = c_Groups_Ozero__class_Ozero(T_c) ) ) ) )).

fof(fact_Suc__diff__1,axiom,(
    ! [V_n] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n))
     => hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),c_Groups_Oone__class_Oone(tc_Nat_Onat))) = V_n ) )).

fof(fact_Max_Oinsert,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( ~ hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
         => ( V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
           => c_Big__Operators_Olinorder__class_OMax(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),V_x_2),c_Big__Operators_Olinorder__class_OMax(T_b,V_A_2)) ) ) ) ) )).

fof(fact_Max_Oinsert__remove,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( ( hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
           => c_Big__Operators_Olinorder__class_OMax(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = V_x_2 )
          & ( hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
           => c_Big__Operators_Olinorder__class_OMax(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),V_x_2),c_Big__Operators_Olinorder__class_OMax(T_b,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))) ) ) ) ) )).

fof(fact_Max_Oeq__fold_H,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( ~ hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
         => c_Big__Operators_Olinorder__class_OMax(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Orderings_Oord__class_Omax(T_b)),V_x_2),V_A_2) ) ) ) )).

fof(fact_even__less__0__iff,axiom,(
    ! [V_a_2,T_b] :
      ( class_Rings_Olinordered__idom(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_a_2),V_a_2)),c_Groups_Ozero__class_Ozero(T_b)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),c_Groups_Ozero__class_Ozero(T_b))) ) ) )).

fof(fact_setsum__mono2,axiom,(
    ! [V_f_2,V_A_2,V_B_2,T_b,T_c] :
      ( class_Groups_Oordered__comm__monoid__add(T_c)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))
         => ( ! [B_b] :
                ( hBOOL(hAPP(hAPP(c_member(T_b),B_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2)))
               => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(V_f_2,B_b))) )
           => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),V_B_2))) ) ) ) ) )).

fof(fact_triple_Osize_I1_J,axiom,(
    ! [V_fun2_2,V_com_2,V_fun1_2,V_fa_2,T_b] : c_Hoare__Mirabelle_Otriple_Otriple__size(T_b,V_fa_2,c_Hoare__Mirabelle_Otriple_Otriple(T_b,V_fun1_2,V_com_2,V_fun2_2)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )).

fof(fact_max__ord__max,axiom,(
    ! [T_b] :
      ( class_Orderings_Oord(T_b)
     => c_Orderings_Oord__class_Omax(T_b) = c_Orderings_Oord_Omax(T_b,c_Orderings_Oord__class_Oless__eq(T_b)) ) )).

fof(fact_Max_Oclosed,axiom,(
    ! [V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
         => ( ! [B_x,B_y] : hBOOL(hAPP(hAPP(c_member(T_b),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),B_x),B_y)),hAPP(hAPP(c_Set_Oinsert(T_b),B_x),hAPP(hAPP(c_Set_Oinsert(T_b),B_y),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))))
           => hBOOL(hAPP(hAPP(c_member(T_b),c_Big__Operators_Olinorder__class_OMax(T_b,V_A_2)),V_A_2)) ) ) ) ) )).

fof(fact_setsum__mono3,axiom,(
    ! [V_f_2,V_A_2,V_B_2,T_b,T_c] :
      ( ( class_Groups_Ocomm__monoid__add(T_c)
        & class_Groups_Oordered__ab__semigroup__add(T_c) )
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))
         => ( ! [B_x] :
                ( hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2)))
               => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(V_f_2,B_x))) )
           => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),V_B_2))) ) ) ) ) )).

fof(fact_min__max_OInf__le__Sup,axiom,(
    ! [V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),V_A_2)),c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),V_A_2))) ) ) ) )).

fof(fact_double__eq__0__iff,axiom,(
    ! [V_a_2,T_b] :
      ( class_Groups_Olinordered__ab__group__add(T_b)
     => ( hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_a_2),V_a_2) = c_Groups_Ozero__class_Ozero(T_b)
      <=> V_a_2 = c_Groups_Ozero__class_Ozero(T_b) ) ) )).

fof(fact_min__max_OSup__fin_Oremove,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
         => ( ( hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
             => c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),V_A_2) = V_x_2 )
            & ( hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
             => c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),V_A_2) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),V_x_2),c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))) ) ) ) ) ) )).

fof(fact_bot__nat__def,axiom,(
    c_Orderings_Obot__class_Obot(tc_Nat_Onat) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )).

fof(fact_max__0R,axiom,(
    ! [V_n] : hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),V_n),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = V_n )).

fof(fact_max__0L,axiom,(
    ! [V_n] : hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n) = V_n )).

fof(fact_min__max_OSup__fin_Oin__idem,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
         => hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),V_x_2),c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),V_A_2)) = c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),V_A_2) ) ) ) )).

fof(fact_min__max_OSup__fin_Osingleton,axiom,(
    ! [V_x_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) = V_x_2 ) )).

fof(fact_min__max_OSup__fin_OF__eq,axiom,(
    ! [V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),V_A_2) = hAPP(c_Finite__Set_Ofold1(T_b,c_Orderings_Oord__class_Omax(T_b)),V_A_2) ) ) )).

fof(fact_min__max_OSup__fin_Oinsert__idem,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
         => c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),V_x_2),c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),V_A_2)) ) ) ) )).

fof(fact_min__max_OSup__fin_Osubset__idem,axiom,(
    ! [V_B_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( V_B_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
         => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2))
           => hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),V_B_2)),c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),V_A_2)) = c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),V_A_2) ) ) ) ) )).

fof(fact_min__max_Oinf__Sup__absorb,axiom,(
    ! [V_a_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2))
         => hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),V_a_2),c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),V_A_2)) = V_a_2 ) ) ) )).

fof(fact_min__max_OSup__fin_Oeq__fold__idem_H,axiom,(
    ! [V_a_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2)) = hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Orderings_Oord__class_Omax(T_b)),V_a_2),V_A_2) ) ) )).

fof(fact_min__max_OSup__fin_Oinsert,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( ~ hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
         => ( V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
           => c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),V_x_2),c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),V_A_2)) ) ) ) ) )).

fof(fact_min__max_OSup__fin_Oinsert__remove,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( ( hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
           => c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = V_x_2 )
          & ( hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
           => c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),V_x_2),c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))) ) ) ) ) )).

fof(fact_min__max_OSup__fin_Oeq__fold_H,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( ~ hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
         => c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Orderings_Oord__class_Omax(T_b)),V_x_2),V_A_2) ) ) ) )).

fof(fact_triple_Osize_I2_J,axiom,(
    ! [V_fun2_2,V_com_2,V_fun1_2,T_b] : hAPP(c_Nat_Osize__class_Osize(tc_Hoare__Mirabelle_Otriple(T_b)),c_Hoare__Mirabelle_Otriple_Otriple(T_b,V_fun1_2,V_com_2,V_fun2_2)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )).

fof(fact_min__max_OSup__fin_Oclosed,axiom,(
    ! [V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
         => ( ! [B_x,B_y] : hBOOL(hAPP(hAPP(c_member(T_b),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),B_x),B_y)),hAPP(hAPP(c_Set_Oinsert(T_b),B_x),hAPP(hAPP(c_Set_Oinsert(T_b),B_y),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))))
           => hBOOL(hAPP(hAPP(c_member(T_b),c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),V_A_2)),V_A_2)) ) ) ) ) )).

fof(fact_card__Suc__eq,axiom,(
    ! [V_k_2,V_A_2,T_b] :
      ( hAPP(c_Finite__Set_Ocard(T_b),V_A_2) = hAPP(c_Nat_OSuc,V_k_2)
    <=> ? [B_b,B_B] :
          ( V_A_2 = hAPP(hAPP(c_Set_Oinsert(T_b),B_b),B_B)
          & ~ hBOOL(hAPP(hAPP(c_member(T_b),B_b),B_B))
          & hAPP(c_Finite__Set_Ocard(T_b),B_B) = V_k_2
          & ( V_k_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
           => B_B = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) ) ) ) )).

fof(fact_Ints__odd__less__0,axiom,(
    ! [V_a_2,T_b] :
      ( class_Rings_Olinordered__idom(T_b)
     => ( hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),c_Int_Oring__1__class_OInts(T_b)))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),c_Groups_Oone__class_Oone(T_b)),V_a_2)),V_a_2)),c_Groups_Ozero__class_Ozero(T_b)))
        <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),c_Groups_Ozero__class_Ozero(T_b))) ) ) ) )).

fof(fact_com_Osize_I4_J,axiom,(
    ! [V_com2,V_com1] : c_Com_Ocom_Ocom__size(c_Com_Ocom_OSemi(V_com1,V_com2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Com_Ocom_Ocom__size(V_com1)),c_Com_Ocom_Ocom__size(V_com2))),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) )).

fof(fact_card__Plus__conv__if,axiom,(
    ! [V_B_2,T_c,V_A_2,T_b] :
      ( ( ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
          & hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) )
       => hAPP(c_Finite__Set_Ocard(tc_sum(T_b,T_c)),c_Sum__Type_OPlus(T_b,T_c,V_A_2,V_B_2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_b),V_A_2)),hAPP(c_Finite__Set_Ocard(T_c),V_B_2)) )
      & ( ~ ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
            & hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) )
       => hAPP(c_Finite__Set_Ocard(tc_sum(T_b,T_c)),c_Sum__Type_OPlus(T_b,T_c,V_A_2,V_B_2)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) ) )).

fof(fact_com_Osize_I12_J,axiom,(
    ! [V_com2,V_com1] : hAPP(c_Nat_Osize__class_Osize(tc_Com_Ocom),c_Com_Ocom_OSemi(V_com1,V_com2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_Com_Ocom),V_com1)),hAPP(c_Nat_Osize__class_Osize(tc_Com_Ocom),V_com2))),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) )).

fof(fact_setprod__diff1,axiom,(
    ! [V_a_2,V_f_2,V_A_2,T_b,T_c] :
      ( class_Fields_Ofield(T_c)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( hAPP(V_f_2,V_a_2) != c_Groups_Ozero__class_Ozero(T_c)
         => ( ( hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2))
             => hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_f_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_f_2),V_A_2)),hAPP(V_f_2,V_a_2)) )
            & ( ~ hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2))
             => hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_f_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_f_2),V_A_2) ) ) ) ) ) )).

fof(fact_divide__zero,axiom,(
    ! [V_a,T_a] :
      ( class_Rings_Odivision__ring__inverse__zero(T_a)
     => hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a)) = c_Groups_Ozero__class_Ozero(T_a) ) )).

fof(fact_divide__zero__left,axiom,(
    ! [V_a,T_a] :
      ( class_Rings_Odivision__ring(T_a)
     => hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a) = c_Groups_Ozero__class_Ozero(T_a) ) )).

fof(fact_add__divide__distrib,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Rings_Odivision__ring(T_a)
     => hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),V_c) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_c)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_b),V_c)) ) )).

fof(fact_times__divide__eq__right,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Rings_Odivision__ring(T_a)
     => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_b),V_c)) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)),V_c) ) )).

fof(fact_diff__divide__distrib,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Rings_Odivision__ring(T_a)
     => hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_b)),V_c) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_c)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_b),V_c)) ) )).

fof(fact_divide__1,axiom,(
    ! [V_a,T_a] :
      ( class_Rings_Odivision__ring(T_a)
     => hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),c_Groups_Oone__class_Oone(T_a)) = V_a ) )).

fof(fact_eq__divide__imp,axiom,(
    ! [V_b,V_a,V_c,T_a] :
      ( class_Rings_Odivision__ring(T_a)
     => ( V_c != c_Groups_Ozero__class_Ozero(T_a)
       => ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c) = V_b
         => V_a = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_b),V_c) ) ) ) )).

fof(fact_divide__eq__imp,axiom,(
    ! [V_a,V_b,V_c,T_a] :
      ( class_Rings_Odivision__ring(T_a)
     => ( V_c != c_Groups_Ozero__class_Ozero(T_a)
       => ( V_b = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)
         => hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_b),V_c) = V_a ) ) ) )).

fof(fact_nonzero__divide__eq__eq,axiom,(
    ! [V_a_2,V_b_2,V_c_2,T_b] :
      ( class_Rings_Odivision__ring(T_b)
     => ( V_c_2 != c_Groups_Ozero__class_Ozero(T_b)
       => ( hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),V_c_2) = V_a_2
        <=> V_b_2 = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_c_2) ) ) ) )).

fof(fact_nonzero__eq__divide__eq,axiom,(
    ! [V_b_2,V_a_2,V_c_2,T_b] :
      ( class_Rings_Odivision__ring(T_b)
     => ( V_c_2 != c_Groups_Ozero__class_Ozero(T_b)
       => ( V_a_2 = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),V_c_2)
        <=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_c_2) = V_b_2 ) ) ) )).

fof(fact_finite__PlusD_I2_J,axiom,(
    ! [V_B_2,V_A_2,T_c,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(tc_sum(T_b,T_c)),c_Sum__Type_OPlus(T_b,T_c,V_A_2,V_B_2)))
     => hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) ) )).

fof(fact_finite__PlusD_I1_J,axiom,(
    ! [V_B_2,V_A_2,T_c,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(tc_sum(T_b,T_c)),c_Sum__Type_OPlus(T_b,T_c,V_A_2,V_B_2)))
     => hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) ) )).

fof(fact_finite__Plus,axiom,(
    ! [V_B_2,T_c,V_A_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2))
       => hBOOL(hAPP(c_Finite__Set_Ofinite(tc_sum(T_b,T_c)),c_Sum__Type_OPlus(T_b,T_c,V_A_2,V_B_2))) ) ) )).

fof(fact_finite__Plus__iff,axiom,(
    ! [V_B_2,V_A_2,T_c,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(tc_sum(T_b,T_c)),c_Sum__Type_OPlus(T_b,T_c,V_A_2,V_B_2)))
    <=> ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
        & hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) ) ) )).

fof(fact_divide__self__if,axiom,(
    ! [V_a,T_a] :
      ( class_Rings_Odivision__ring__inverse__zero(T_a)
     => ( ( V_a = c_Groups_Ozero__class_Ozero(T_a)
         => hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_a) = c_Groups_Ozero__class_Ozero(T_a) )
        & ( V_a != c_Groups_Ozero__class_Ozero(T_a)
         => hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_a) = c_Groups_Oone__class_Oone(T_a) ) ) ) )).

fof(fact_divide__self,axiom,(
    ! [V_a,T_a] :
      ( class_Rings_Odivision__ring(T_a)
     => ( V_a != c_Groups_Ozero__class_Ozero(T_a)
       => hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_a) = c_Groups_Oone__class_Oone(T_a) ) ) )).

fof(fact_right__inverse__eq,axiom,(
    ! [V_a_2,V_b_2,T_b] :
      ( class_Rings_Odivision__ring(T_b)
     => ( V_b_2 != c_Groups_Ozero__class_Ozero(T_b)
       => ( hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_a_2),V_b_2) = c_Groups_Oone__class_Oone(T_b)
        <=> V_a_2 = V_b_2 ) ) ) )).

fof(fact_Ints__0,axiom,(
    ! [T_b] :
      ( class_Rings_Oring__1(T_b)
     => hBOOL(hAPP(hAPP(c_member(T_b),c_Groups_Ozero__class_Ozero(T_b)),c_Int_Oring__1__class_OInts(T_b))) ) )).

fof(fact_Ints__add,axiom,(
    ! [V_b_2,V_a_2,T_b] :
      ( class_Rings_Oring__1(T_b)
     => ( hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),c_Int_Oring__1__class_OInts(T_b)))
       => ( hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),c_Int_Oring__1__class_OInts(T_b)))
         => hBOOL(hAPP(hAPP(c_member(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_a_2),V_b_2)),c_Int_Oring__1__class_OInts(T_b))) ) ) ) )).

fof(fact_Ints__mult,axiom,(
    ! [V_b_2,V_a_2,T_b] :
      ( class_Rings_Oring__1(T_b)
     => ( hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),c_Int_Oring__1__class_OInts(T_b)))
       => ( hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),c_Int_Oring__1__class_OInts(T_b)))
         => hBOOL(hAPP(hAPP(c_member(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_b_2)),c_Int_Oring__1__class_OInts(T_b))) ) ) ) )).

fof(fact_Ints__diff,axiom,(
    ! [V_b_2,V_a_2,T_b] :
      ( class_Rings_Oring__1(T_b)
     => ( hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),c_Int_Oring__1__class_OInts(T_b)))
       => ( hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),c_Int_Oring__1__class_OInts(T_b)))
         => hBOOL(hAPP(hAPP(c_member(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),V_a_2),V_b_2)),c_Int_Oring__1__class_OInts(T_b))) ) ) ) )).

fof(fact_Ints__1,axiom,(
    ! [T_b] :
      ( class_Rings_Oring__1(T_b)
     => hBOOL(hAPP(hAPP(c_member(T_b),c_Groups_Oone__class_Oone(T_b)),c_Int_Oring__1__class_OInts(T_b))) ) )).

fof(fact_com_Osize_I9_J,axiom,(
    hAPP(c_Nat_Osize__class_Osize(tc_Com_Ocom),c_Com_Ocom_OSKIP) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )).

fof(fact_com_Osize_I1_J,axiom,(
    c_Com_Ocom_Ocom__size(c_Com_Ocom_OSKIP) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )).

fof(fact_Ints__double__eq__0__iff,axiom,(
    ! [V_a_2,T_b] :
      ( class_Int_Oring__char__0(T_b)
     => ( hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),c_Int_Oring__1__class_OInts(T_b)))
       => ( hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_a_2),V_a_2) = c_Groups_Ozero__class_Ozero(T_b)
        <=> V_a_2 = c_Groups_Ozero__class_Ozero(T_b) ) ) ) )).

fof(fact_card__Plus,axiom,(
    ! [V_B_2,T_c,V_A_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2))
       => hAPP(c_Finite__Set_Ocard(tc_sum(T_b,T_c)),c_Sum__Type_OPlus(T_b,T_c,V_A_2,V_B_2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_b),V_A_2)),hAPP(c_Finite__Set_Ocard(T_c),V_B_2)) ) ) )).

fof(fact_Ints__odd__nonzero,axiom,(
    ! [V_a_2,T_b] :
      ( class_Int_Oring__char__0(T_b)
     => ( hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),c_Int_Oring__1__class_OInts(T_b)))
       => hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),c_Groups_Oone__class_Oone(T_b)),V_a_2)),V_a_2) != c_Groups_Ozero__class_Ozero(T_b) ) ) )).

fof(fact_divide__left__mono__neg,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Fields_Olinordered__field__inverse__zero(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_c),c_Groups_Ozero__class_Ozero(T_a)))
         => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)))
           => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_c),V_a)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_c),V_b))) ) ) ) ) )).

fof(fact_divide__left__mono,axiom,(
    ! [V_c,V_a,V_b,T_a] :
      ( class_Fields_Olinordered__field(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),V_a))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c))
         => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)))
           => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_c),V_a)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_c),V_b))) ) ) ) ) )).

fof(fact_neg__divide__le__eq,axiom,(
    ! [V_a_2,V_b_2,V_c_2,T_b] :
      ( class_Fields_Olinordered__field(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),c_Groups_Ozero__class_Ozero(T_b)))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),V_c_2)),V_a_2))
        <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_c_2)),V_b_2)) ) ) ) )).

fof(fact_neg__le__divide__eq,axiom,(
    ! [V_b_2,V_a_2,V_c_2,T_b] :
      ( class_Fields_Olinordered__field(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),c_Groups_Ozero__class_Ozero(T_b)))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),V_c_2)))
        <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_c_2))) ) ) ) )).

fof(fact_mult__imp__le__div__pos,axiom,(
    ! [V_x,V_z,V_y,T_a] :
      ( class_Fields_Olinordered__field(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_y))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_z),V_y)),V_x))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_z),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_y))) ) ) ) )).

fof(fact_mult__imp__div__pos__le,axiom,(
    ! [V_z,V_x,V_y,T_a] :
      ( class_Fields_Olinordered__field(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_y))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_z),V_y)))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_y)),V_z)) ) ) ) )).

fof(fact_pos__divide__le__eq,axiom,(
    ! [V_a_2,V_b_2,V_c_2,T_b] :
      ( class_Fields_Olinordered__field(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_c_2))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),V_c_2)),V_a_2))
        <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_c_2))) ) ) ) )).

fof(fact_pos__le__divide__eq,axiom,(
    ! [V_b_2,V_a_2,V_c_2,T_b] :
      ( class_Fields_Olinordered__field(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_c_2))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),V_c_2)))
        <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_c_2)),V_b_2)) ) ) ) )).

fof(fact_times__divide__times__eq,axiom,(
    ! [V_w,V_z,V_y,V_x,T_a] :
      ( class_Fields_Ofield__inverse__zero(T_a)
     => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_y)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_z),V_w)) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_z)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_y),V_w)) ) )).

fof(fact_zero__le__divide__iff,axiom,(
    ! [V_b_2,V_a_2,T_b] :
      ( class_Fields_Olinordered__field__inverse__zero(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_a_2),V_b_2)))
      <=> ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_a_2))
            & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_b_2)) )
          | ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),c_Groups_Ozero__class_Ozero(T_b)))
            & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_b_2),c_Groups_Ozero__class_Ozero(T_b))) ) ) ) ) )).

fof(fact_divide__le__0__iff,axiom,(
    ! [V_b_2,V_a_2,T_b] :
      ( class_Fields_Olinordered__field__inverse__zero(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_a_2),V_b_2)),c_Groups_Ozero__class_Ozero(T_b)))
      <=> ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_a_2))
            & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_b_2),c_Groups_Ozero__class_Ozero(T_b))) )
          | ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),c_Groups_Ozero__class_Ozero(T_b)))
            & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_b_2)) ) ) ) ) )).

fof(fact_divide__right__mono,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Fields_Olinordered__field__inverse__zero(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_c)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_b),V_c))) ) ) ) )).

fof(fact_divide__right__mono__neg,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Fields_Olinordered__field__inverse__zero(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_c),c_Groups_Ozero__class_Ozero(T_a)))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_b),V_c)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_c))) ) ) ) )).

fof(fact_zero__less__divide__iff,axiom,(
    ! [V_b_2,V_a_2,T_b] :
      ( class_Fields_Olinordered__field__inverse__zero(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_a_2),V_b_2)))
      <=> ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_a_2))
            & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_b_2)) )
          | ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),c_Groups_Ozero__class_Ozero(T_b)))
            & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_b_2),c_Groups_Ozero__class_Ozero(T_b))) ) ) ) ) )).

fof(fact_divide__less__0__iff,axiom,(
    ! [V_b_2,V_a_2,T_b] :
      ( class_Fields_Olinordered__field__inverse__zero(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_a_2),V_b_2)),c_Groups_Ozero__class_Ozero(T_b)))
      <=> ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_a_2))
            & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_b_2),c_Groups_Ozero__class_Ozero(T_b))) )
          | ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),c_Groups_Ozero__class_Ozero(T_b)))
            & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_b_2)) ) ) ) ) )).

fof(fact_divide__pos__pos,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Fields_Olinordered__field(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_x))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_y))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_y))) ) ) ) )).

fof(fact_divide__pos__neg,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Fields_Olinordered__field(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_x))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),c_Groups_Ozero__class_Ozero(T_a)))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_y)),c_Groups_Ozero__class_Ozero(T_a))) ) ) ) )).

fof(fact_divide__neg__pos,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Fields_Olinordered__field(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),c_Groups_Ozero__class_Ozero(T_a)))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_y))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_y)),c_Groups_Ozero__class_Ozero(T_a))) ) ) ) )).

fof(fact_divide__neg__neg,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Fields_Olinordered__field(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),c_Groups_Ozero__class_Ozero(T_a)))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),c_Groups_Ozero__class_Ozero(T_a)))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_y))) ) ) ) )).

fof(fact_divide__strict__right__mono,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Fields_Olinordered__field(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_c)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_b),V_c))) ) ) ) )).

fof(fact_divide__strict__right__mono__neg,axiom,(
    ! [V_c,V_a,V_b,T_a] :
      ( class_Fields_Olinordered__field(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),V_a))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_c),c_Groups_Ozero__class_Ozero(T_a)))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_c)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_b),V_c))) ) ) ) )).

fof(fact_frac__eq__eq,axiom,(
    ! [V_w_2,V_x_2,V_z_2,V_y_2,T_b] :
      ( class_Fields_Ofield(T_b)
     => ( V_y_2 != c_Groups_Ozero__class_Ozero(T_b)
       => ( V_z_2 != c_Groups_Ozero__class_Ozero(T_b)
         => ( hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_x_2),V_y_2) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_w_2),V_z_2)
          <=> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_x_2),V_z_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_w_2),V_y_2) ) ) ) ) )).

fof(fact_mult__divide__mult__cancel__left,axiom,(
    ! [V_b,V_a,V_c,T_a] :
      ( class_Fields_Ofield__inverse__zero(T_a)
     => ( V_c != c_Groups_Ozero__class_Ozero(T_a)
       => hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b)) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_b) ) ) )).

fof(fact_mult__divide__mult__cancel__right,axiom,(
    ! [V_b,V_a,V_c,T_a] :
      ( class_Fields_Ofield__inverse__zero(T_a)
     => ( V_c != c_Groups_Ozero__class_Ozero(T_a)
       => hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c)) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_b) ) ) )).

fof(fact_divide__eq__eq,axiom,(
    ! [V_a_2,V_c_2,V_b_2,T_b] :
      ( class_Fields_Ofield__inverse__zero(T_b)
     => ( hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),V_c_2) = V_a_2
      <=> ( ( V_c_2 != c_Groups_Ozero__class_Ozero(T_b)
           => V_b_2 = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_c_2) )
          & ( V_c_2 = c_Groups_Ozero__class_Ozero(T_b)
           => V_a_2 = c_Groups_Ozero__class_Ozero(T_b) ) ) ) ) )).

fof(fact_eq__divide__eq,axiom,(
    ! [V_c_2,V_b_2,V_a_2,T_b] :
      ( class_Fields_Ofield__inverse__zero(T_b)
     => ( V_a_2 = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),V_c_2)
      <=> ( ( V_c_2 != c_Groups_Ozero__class_Ozero(T_b)
           => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_c_2) = V_b_2 )
          & ( V_c_2 = c_Groups_Ozero__class_Ozero(T_b)
           => V_a_2 = c_Groups_Ozero__class_Ozero(T_b) ) ) ) ) )).

fof(fact_divide__nonpos__neg,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Fields_Olinordered__field(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),c_Groups_Ozero__class_Ozero(T_a)))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),c_Groups_Ozero__class_Ozero(T_a)))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_y))) ) ) ) )).

fof(fact_divide__nonpos__pos,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Fields_Olinordered__field(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),c_Groups_Ozero__class_Ozero(T_a)))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_y))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_y)),c_Groups_Ozero__class_Ozero(T_a))) ) ) ) )).

fof(fact_frac__le,axiom,(
    ! [V_z,V_w,V_y,V_x,T_a] :
      ( class_Fields_Olinordered__field(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_x))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y))
         => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_w))
           => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_w),V_z))
             => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_z)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_y),V_w))) ) ) ) ) ) )).

fof(fact_frac__less,axiom,(
    ! [V_z,V_w,V_y,V_x,T_a] :
      ( class_Fields_Olinordered__field(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_x))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y))
         => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_w))
           => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_w),V_z))
             => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_z)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_y),V_w))) ) ) ) ) ) )).

fof(fact_frac__less2,axiom,(
    ! [V_z,V_w,V_y,V_x,T_a] :
      ( class_Fields_Olinordered__field(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_x))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y))
         => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_w))
           => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_w),V_z))
             => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_z)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_y),V_w))) ) ) ) ) ) )).

fof(fact_divide__nonneg__neg,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Fields_Olinordered__field(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_x))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),c_Groups_Ozero__class_Ozero(T_a)))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_y)),c_Groups_Ozero__class_Ozero(T_a))) ) ) ) )).

fof(fact_divide__nonneg__pos,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Fields_Olinordered__field(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_x))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_y))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_y))) ) ) ) )).

fof(fact_less__divide__eq,axiom,(
    ! [V_c_2,V_b_2,V_a_2,T_b] :
      ( class_Fields_Olinordered__field__inverse__zero(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),V_c_2)))
      <=> ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_c_2))
           => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_c_2)),V_b_2)) )
          & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_c_2))
           => ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),c_Groups_Ozero__class_Ozero(T_b)))
               => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_c_2))) )
              & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),c_Groups_Ozero__class_Ozero(T_b)))
               => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),c_Groups_Ozero__class_Ozero(T_b))) ) ) ) ) ) ) )).

fof(fact_divide__less__eq,axiom,(
    ! [V_a_2,V_c_2,V_b_2,T_b] :
      ( class_Fields_Olinordered__field__inverse__zero(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),V_c_2)),V_a_2))
      <=> ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_c_2))
           => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_c_2))) )
          & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_c_2))
           => ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),c_Groups_Ozero__class_Ozero(T_b)))
               => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_c_2)),V_b_2)) )
              & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),c_Groups_Ozero__class_Ozero(T_b)))
               => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_a_2)) ) ) ) ) ) ) )).

fof(fact_pos__less__divide__eq,axiom,(
    ! [V_b_2,V_a_2,V_c_2,T_b] :
      ( class_Fields_Olinordered__field(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_c_2))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),V_c_2)))
        <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_c_2)),V_b_2)) ) ) ) )).

fof(fact_pos__divide__less__eq,axiom,(
    ! [V_a_2,V_b_2,V_c_2,T_b] :
      ( class_Fields_Olinordered__field(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_c_2))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),V_c_2)),V_a_2))
        <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_c_2))) ) ) ) )).

fof(fact_mult__imp__div__pos__less,axiom,(
    ! [V_z,V_x,V_y,T_a] :
      ( class_Fields_Olinordered__field(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_y))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_z),V_y)))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_y)),V_z)) ) ) ) )).

fof(fact_mult__imp__less__div__pos,axiom,(
    ! [V_x,V_z,V_y,T_a] :
      ( class_Fields_Olinordered__field(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_y))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_z),V_y)),V_x))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_z),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_y))) ) ) ) )).

fof(fact_neg__less__divide__eq,axiom,(
    ! [V_b_2,V_a_2,V_c_2,T_b] :
      ( class_Fields_Olinordered__field(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),c_Groups_Ozero__class_Ozero(T_b)))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),V_c_2)))
        <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_c_2))) ) ) ) )).

fof(fact_neg__divide__less__eq,axiom,(
    ! [V_a_2,V_b_2,V_c_2,T_b] :
      ( class_Fields_Olinordered__field(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),c_Groups_Ozero__class_Ozero(T_b)))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),V_c_2)),V_a_2))
        <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_c_2)),V_b_2)) ) ) ) )).

fof(fact_divide__strict__left__mono,axiom,(
    ! [V_c,V_a,V_b,T_a] :
      ( class_Fields_Olinordered__field(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),V_a))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c))
         => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)))
           => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_c),V_a)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_c),V_b))) ) ) ) ) )).

fof(fact_divide__strict__left__mono__neg,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Fields_Olinordered__field(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_c),c_Groups_Ozero__class_Ozero(T_a)))
         => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)))
           => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_c),V_a)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_c),V_b))) ) ) ) ) )).

fof(fact_add__num__frac,axiom,(
    ! [V_x,V_z,V_y,T_a] :
      ( class_Fields_Ofield__inverse__zero(T_a)
     => ( V_y != c_Groups_Ozero__class_Ozero(T_a)
       => hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_z),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_y)) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_z),V_y))),V_y) ) ) )).

fof(fact_add__divide__eq__iff,axiom,(
    ! [V_y,V_x,V_z,T_a] :
      ( class_Fields_Ofield(T_a)
     => ( V_z != c_Groups_Ozero__class_Ozero(T_a)
       => hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_x),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_y),V_z)) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_z),V_x)),V_y)),V_z) ) ) )).

fof(fact_add__frac__num,axiom,(
    ! [V_z,V_x,V_y,T_a] :
      ( class_Fields_Ofield__inverse__zero(T_a)
     => ( V_y != c_Groups_Ozero__class_Ozero(T_a)
       => hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_y)),V_z) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_z),V_y))),V_y) ) ) )).

fof(fact_divide__add__eq__iff,axiom,(
    ! [V_y,V_x,V_z,T_a] :
      ( class_Fields_Ofield(T_a)
     => ( V_z != c_Groups_Ozero__class_Ozero(T_a)
       => hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_z)),V_y) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_z),V_y))),V_z) ) ) )).

fof(fact_add__frac__eq,axiom,(
    ! [V_w,V_x,V_z,V_y,T_a] :
      ( class_Fields_Ofield(T_a)
     => ( V_y != c_Groups_Ozero__class_Ozero(T_a)
       => ( V_z != c_Groups_Ozero__class_Ozero(T_a)
         => hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_y)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_w),V_z)) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_z)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_w),V_y))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_y),V_z)) ) ) ) )).

fof(fact_diff__divide__eq__iff,axiom,(
    ! [V_y,V_x,V_z,T_a] :
      ( class_Fields_Ofield(T_a)
     => ( V_z != c_Groups_Ozero__class_Ozero(T_a)
       => hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_x),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_y),V_z)) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_z),V_x)),V_y)),V_z) ) ) )).

fof(fact_divide__diff__eq__iff,axiom,(
    ! [V_y,V_x,V_z,T_a] :
      ( class_Fields_Ofield(T_a)
     => ( V_z != c_Groups_Ozero__class_Ozero(T_a)
       => hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_z)),V_y) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_z),V_y))),V_z) ) ) )).

fof(fact_diff__frac__eq,axiom,(
    ! [V_w,V_x,V_z,V_y,T_a] :
      ( class_Fields_Ofield(T_a)
     => ( V_y != c_Groups_Ozero__class_Ozero(T_a)
       => ( V_z != c_Groups_Ozero__class_Ozero(T_a)
         => hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_y)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_w),V_z)) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_z)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_w),V_y))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_y),V_z)) ) ) ) )).

fof(fact_gt__half__sum,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Fields_Olinordered__field(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Groups_Oone__class_Oone(T_a)),c_Groups_Oone__class_Oone(T_a)))),V_b)) ) ) )).

fof(fact_less__half__sum,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Fields_Olinordered__field(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Groups_Oone__class_Oone(T_a)),c_Groups_Oone__class_Oone(T_a))))) ) ) )).

fof(fact_le__divide__eq,axiom,(
    ! [V_c_2,V_b_2,V_a_2,T_b] :
      ( class_Fields_Olinordered__field__inverse__zero(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),V_c_2)))
      <=> ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_c_2))
           => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_c_2)),V_b_2)) )
          & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_c_2))
           => ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),c_Groups_Ozero__class_Ozero(T_b)))
               => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_c_2))) )
              & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),c_Groups_Ozero__class_Ozero(T_b)))
               => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),c_Groups_Ozero__class_Ozero(T_b))) ) ) ) ) ) ) )).

fof(fact_divide__le__eq,axiom,(
    ! [V_a_2,V_c_2,V_b_2,T_b] :
      ( class_Fields_Olinordered__field__inverse__zero(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),V_c_2)),V_a_2))
      <=> ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_c_2))
           => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_c_2))) )
          & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_c_2))
           => ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),c_Groups_Ozero__class_Ozero(T_b)))
               => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_c_2)),V_b_2)) )
              & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),c_Groups_Ozero__class_Ozero(T_b)))
               => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_a_2)) ) ) ) ) ) ) )).

fof(fact_Plus__eq__empty__conv,axiom,(
    ! [V_B_2,V_A_2,T_c,T_b] :
      ( c_Sum__Type_OPlus(T_b,T_c,V_A_2,V_B_2) = c_Orderings_Obot__class_Obot(tc_fun(tc_sum(T_b,T_c),tc_HOL_Obool))
    <=> ( V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
        & V_B_2 = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) ) ) )).

fof(fact_ex__least__nat__less,axiom,(
    ! [V_n_2,V_P_2] :
      ( ~ hBOOL(hAPP(V_P_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))
     => ( hBOOL(hAPP(V_P_2,V_n_2))
       => ? [B_k] :
            ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_k),V_n_2))
            & ! [B_i] :
                ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B_i),B_k))
               => ~ hBOOL(hAPP(V_P_2,B_i)) )
            & hBOOL(hAPP(V_P_2,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B_k),c_Groups_Oone__class_Oone(tc_Nat_Onat)))) ) ) ) )).

fof(fact_setsum__mono__zero__left,axiom,(
    ! [V_f_2,V_S_2,V_T_2,T_b,T_c] :
      ( class_Groups_Ocomm__monoid__add(T_c)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_T_2))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_S_2),V_T_2))
         => ( ! [B_x] :
                ( hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_T_2),V_S_2)))
               => hAPP(V_f_2,B_x) = c_Groups_Ozero__class_Ozero(T_c) )
           => hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),V_S_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),V_T_2) ) ) ) ) )).

fof(fact_setsum__mono__zero__right,axiom,(
    ! [V_f_2,V_S_2,V_T_2,T_b,T_c] :
      ( class_Groups_Ocomm__monoid__add(T_c)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_T_2))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_S_2),V_T_2))
         => ( ! [B_x] :
                ( hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_T_2),V_S_2)))
               => hAPP(V_f_2,B_x) = c_Groups_Ozero__class_Ozero(T_c) )
           => hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),V_T_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),V_S_2) ) ) ) ) )).

fof(fact_field__le__mult__one__interval,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Fields_Olinordered__field__inverse__zero(T_a)
     => ( ! [B_z] :
            ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),B_z))
           => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),B_z),c_Groups_Oone__class_Oone(T_a)))
             => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),B_z),V_x)),V_y)) ) )
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y)) ) ) )).

fof(fact_setprod__pos__nat,axiom,(
    ! [V_f_2,V_S_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_S_2))
     => ( ! [B_x] :
            ( hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_S_2))
           => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(V_f_2,B_x))) )
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,tc_Nat_Onat),V_f_2),V_S_2))) ) ) )).

fof(fact_setsum__SucD,axiom,(
    ! [V_n_2,V_A_2,V_f_2,T_b] :
      ( hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,tc_Nat_Onat),V_f_2),V_A_2) = hAPP(c_Nat_OSuc,V_n_2)
     => ? [B_x] :
          ( hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2))
          & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(V_f_2,B_x))) ) ) )).

fof(fact_com_Osize_I5_J,axiom,(
    ! [V_com2_2,V_com1_2,V_fun_2] : c_Com_Ocom_Ocom__size(c_Com_Ocom_OCond(V_fun_2,V_com1_2,V_com2_2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Com_Ocom_Ocom__size(V_com1_2)),c_Com_Ocom_Ocom__size(V_com2_2))),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) )).

fof(fact_com_Osimps_I4_J,axiom,(
    ! [V_com2_H_2,V_com1_H_2,V_fun_H_2,V_com2_2,V_com1_2,V_fun_2] :
      ( c_Com_Ocom_OCond(V_fun_2,V_com1_2,V_com2_2) = c_Com_Ocom_OCond(V_fun_H_2,V_com1_H_2,V_com2_H_2)
    <=> ( V_fun_2 = V_fun_H_2
        & V_com1_2 = V_com1_H_2
        & V_com2_2 = V_com2_H_2 ) ) )).

fof(fact_com_Osimps_I44_J,axiom,(
    ! [V_com2_H_2,V_com1_H_2,V_fun_H_2,V_com2_2,V_com1_2] : c_Com_Ocom_OSemi(V_com1_2,V_com2_2) != c_Com_Ocom_OCond(V_fun_H_2,V_com1_H_2,V_com2_H_2) )).

fof(fact_com_Osimps_I45_J,axiom,(
    ! [V_com2_2,V_com1_2,V_com2_H_2,V_com1_H_2,V_fun_H_2] : c_Com_Ocom_OCond(V_fun_H_2,V_com1_H_2,V_com2_H_2) != c_Com_Ocom_OSemi(V_com1_2,V_com2_2) )).

fof(fact_com_Osimps_I14_J,axiom,(
    ! [V_com2_H_2,V_com1_H_2,V_fun_H_2] : c_Com_Ocom_OSKIP != c_Com_Ocom_OCond(V_fun_H_2,V_com1_H_2,V_com2_H_2) )).

fof(fact_com_Osimps_I15_J,axiom,(
    ! [V_com2_H_2,V_com1_H_2,V_fun_H_2] : c_Com_Ocom_OCond(V_fun_H_2,V_com1_H_2,V_com2_H_2) != c_Com_Ocom_OSKIP )).

fof(fact_com_Osize_I13_J,axiom,(
    ! [V_com2_2,V_com1_2,V_fun_2] : hAPP(c_Nat_Osize__class_Osize(tc_Com_Ocom),c_Com_Ocom_OCond(V_fun_2,V_com1_2,V_com2_2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_Com_Ocom),V_com1_2)),hAPP(c_Nat_Osize__class_Osize(tc_Com_Ocom),V_com2_2))),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) )).

fof(fact_card_Oneutral,axiom,(
    ! [V_A_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
     => ( ( ? [B_x] : hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2))
         => c_Groups_Oone__class_Oone(tc_Nat_Onat) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )
       => hAPP(c_Finite__Set_Ocard(T_b),V_A_2) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) ) )).

fof(fact_min__max_Ofold__sup__le__sup,axiom,(
    ! [V_c_2,V_b_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( ! [B_x] :
              ( hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2))
             => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),B_x),V_b_2)) )
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Orderings_Oord__class_Omax(T_b)),V_c_2),V_A_2)),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),V_b_2),V_c_2))) ) ) ) )).

fof(fact_min__max_Oinf__le__fold__inf,axiom,(
    ! [V_c_2,V_b_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( ! [B_x] :
              ( hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2))
             => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_b_2),B_x)) )
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),V_b_2),V_c_2)),hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Orderings_Oord__class_Omin(T_b)),V_c_2),V_A_2))) ) ) ) )).

fof(fact_Max__eqI,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( ! [B_y] :
              ( hBOOL(hAPP(hAPP(c_member(T_b),B_y),V_A_2))
             => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),B_y),V_x_2)) )
         => ( hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
           => c_Big__Operators_Olinorder__class_OMax(T_b,V_A_2) = V_x_2 ) ) ) ) )).

fof(fact_Min__eqI,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( ! [B_y] :
              ( hBOOL(hAPP(hAPP(c_member(T_b),B_y),V_A_2))
             => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),B_y)) )
         => ( hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
           => c_Big__Operators_Olinorder__class_OMin(T_b,V_A_2) = V_x_2 ) ) ) ) )).

fof(fact_expand__Suc,axiom,(
    ! [V_v] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)))
     => hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v) = hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),c_Groups_Oone__class_Oone(tc_Nat_Onat))) ) )).

fof(fact_com_Osize_I11_J,axiom,(
    ! [V_com_2,V_fun_2,V_loc_2] : hAPP(c_Nat_Osize__class_Osize(tc_Com_Ocom),c_Com_Ocom_OLocal(V_loc_2,V_fun_2,V_com_2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_Com_Ocom),V_com_2)),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) )).

fof(fact_com_Osize_I3_J,axiom,(
    ! [V_com_2,V_fun_2,V_loc_2] : c_Com_Ocom_Ocom__size(c_Com_Ocom_OLocal(V_loc_2,V_fun_2,V_com_2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Com_Ocom_Ocom__size(V_com_2)),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) )).

fof(fact_com_Osimps_I2_J,axiom,(
    ! [V_com_H_2,V_fun_H_2,V_loc_H_2,V_com_2,V_fun_2,V_loc_2] :
      ( c_Com_Ocom_OLocal(V_loc_2,V_fun_2,V_com_2) = c_Com_Ocom_OLocal(V_loc_H_2,V_fun_H_2,V_com_H_2)
    <=> ( V_loc_2 = V_loc_H_2
        & V_fun_2 = V_fun_H_2
        & V_com_2 = V_com_H_2 ) ) )).

fof(fact_number__of__reorient,axiom,(
    ! [V_x_2,V_w_2,T_b] :
      ( class_Int_Onumber(T_b)
     => ( hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2) = V_x_2
      <=> V_x_2 = hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2) ) ) )).

fof(fact_eq__number__of,axiom,(
    ! [V_y_2,V_x_2,T_b] :
      ( ( class_Int_Onumber__ring(T_b)
        & class_Int_Oring__char__0(T_b) )
     => ( hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_x_2) = hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_y_2)
      <=> V_x_2 = V_y_2 ) ) )).

fof(fact_le__number__of,axiom,(
    ! [V_y_2,V_x_2,T_b] :
      ( ( class_Int_Onumber__ring(T_b)
        & class_Rings_Olinordered__idom(T_b) )
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_x_2)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_y_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_x_2),V_y_2)) ) ) )).

fof(fact_less__number__of,axiom,(
    ! [V_y_2,V_x_2,T_b] :
      ( ( class_Int_Onumber__ring(T_b)
        & class_Rings_Olinordered__idom(T_b) )
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_x_2)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_y_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_x_2),V_y_2)) ) ) )).

fof(fact_number__of__add,axiom,(
    ! [V_w,V_v,T_a] :
      ( class_Int_Onumber__ring(T_a)
     => hAPP(c_Int_Onumber__class_Onumber__of(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_v),V_w)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w)) ) )).

fof(fact_add__number__of__eq,axiom,(
    ! [V_w,V_v,T_a] :
      ( class_Int_Onumber__ring(T_a)
     => hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w)) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_v),V_w)) ) )).

fof(fact_add__number__of__left,axiom,(
    ! [V_z,V_w,V_v,T_a] :
      ( class_Int_Onumber__ring(T_a)
     => hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_v)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w)),V_z)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_v),V_w))),V_z) ) )).

fof(fact_number__of__mult,axiom,(
    ! [V_w,V_v,T_a] :
      ( class_Int_Onumber__ring(T_a)
     => hAPP(c_Int_Onumber__class_Onumber__of(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_v),V_w)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w)) ) )).

fof(fact_arith__simps_I32_J,axiom,(
    ! [V_w,V_v,T_a] :
      ( class_Int_Onumber__ring(T_a)
     => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w)) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_v),V_w)) ) )).

fof(fact_mult__number__of__left,axiom,(
    ! [V_z,V_w,V_v,T_a] :
      ( class_Int_Onumber__ring(T_a)
     => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_v)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w)),V_z)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_v),V_w))),V_z) ) )).

fof(fact_number__of__diff,axiom,(
    ! [V_w,V_v,T_a] :
      ( class_Int_Onumber__ring(T_a)
     => hAPP(c_Int_Onumber__class_Onumber__of(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_v),V_w)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w)) ) )).

fof(fact_com_Osimps_I37_J,axiom,(
    ! [V_com_2,V_fun_2,V_loc_2,V_com2_H_2,V_com1_H_2,V_fun_H_2] : c_Com_Ocom_OCond(V_fun_H_2,V_com1_H_2,V_com2_H_2) != c_Com_Ocom_OLocal(V_loc_2,V_fun_2,V_com_2) )).

fof(fact_com_Osimps_I36_J,axiom,(
    ! [V_com2_H_2,V_com1_H_2,V_fun_H_2,V_com_2,V_fun_2,V_loc_2] : c_Com_Ocom_OLocal(V_loc_2,V_fun_2,V_com_2) != c_Com_Ocom_OCond(V_fun_H_2,V_com1_H_2,V_com2_H_2) )).

fof(fact_com_Osimps_I34_J,axiom,(
    ! [V_com2_H_2,V_com1_H_2,V_com_2,V_fun_2,V_loc_2] : c_Com_Ocom_OLocal(V_loc_2,V_fun_2,V_com_2) != c_Com_Ocom_OSemi(V_com1_H_2,V_com2_H_2) )).

fof(fact_com_Osimps_I35_J,axiom,(
    ! [V_com_2,V_fun_2,V_loc_2,V_com2_H_2,V_com1_H_2] : c_Com_Ocom_OSemi(V_com1_H_2,V_com2_H_2) != c_Com_Ocom_OLocal(V_loc_2,V_fun_2,V_com_2) )).

fof(fact_com_Osimps_I11_J,axiom,(
    ! [V_com_H_2,V_fun_H_2,V_loc_H_2] : c_Com_Ocom_OLocal(V_loc_H_2,V_fun_H_2,V_com_H_2) != c_Com_Ocom_OSKIP )).

fof(fact_com_Osimps_I10_J,axiom,(
    ! [V_com_H_2,V_fun_H_2,V_loc_H_2] : c_Com_Ocom_OSKIP != c_Com_Ocom_OLocal(V_loc_H_2,V_fun_H_2,V_com_H_2) )).

fof(fact_le__number__of__eq__not__less,axiom,(
    ! [V_w_2,V_v_2,T_b] :
      ( ( class_Int_Onumber(T_b)
        & class_Orderings_Olinorder(T_b) )
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v_2)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)))
      <=> ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v_2))) ) ) )).

fof(fact_right__distrib__number__of,axiom,(
    ! [V_c,V_b,V_v,T_b] :
      ( ( class_Int_Onumber(T_b)
        & class_Rings_Osemiring(T_b) )
     => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_b),V_c)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v)),V_b)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v)),V_c)) ) )).

fof(fact_left__distrib__number__of,axiom,(
    ! [V_v,V_b,V_a,T_b] :
      ( ( class_Int_Onumber(T_b)
        & class_Rings_Osemiring(T_b) )
     => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_a),V_b)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v))) ) )).

fof(fact_add__number__of__diff1,axiom,(
    ! [V_c,V_w,V_v,T_a] :
      ( class_Int_Onumber__ring(T_a)
     => hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_v)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w)),V_c)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_v),V_w))),V_c) ) )).

fof(fact_right__diff__distrib__number__of,axiom,(
    ! [V_c,V_b,V_v,T_b] :
      ( ( class_Int_Onumber(T_b)
        & class_Rings_Oring(T_b) )
     => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),V_b),V_c)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v)),V_b)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v)),V_c)) ) )).

fof(fact_left__diff__distrib__number__of,axiom,(
    ! [V_v,V_b,V_a,T_b] :
      ( ( class_Int_Onumber(T_b)
        & class_Rings_Oring(T_b) )
     => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),V_a),V_b)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v))) ) )).

fof(fact_min__number__of,axiom,(
    ! [V_v,V_u,T_b] :
      ( ( class_Int_Onumber(T_b)
        & class_Orderings_Oord(T_b) )
     => ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_u)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v)))
         => hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_u)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v)) = hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_u) )
        & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_u)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v)))
         => hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_u)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v)) = hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v) ) ) ) )).

fof(fact_max__number__of,axiom,(
    ! [V_v,V_u,T_b] :
      ( ( class_Int_Onumber(T_b)
        & class_Orderings_Oord(T_b) )
     => ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_u)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v)))
         => hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_u)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v)) = hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v) )
        & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_u)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v)))
         => hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_u)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v)) = hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_u) ) ) ) )).

fof(fact_Ints__number__of,axiom,(
    ! [V_w_2,T_b] :
      ( class_Int_Onumber__ring(T_b)
     => hBOOL(hAPP(hAPP(c_member(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)),c_Int_Oring__1__class_OInts(T_b))) ) )).

fof(fact_divide__eq__eq__number__of1,axiom,(
    ! [V_a_2,V_w_2,V_b_2,T_b] :
      ( ( class_Fields_Ofield__inverse__zero(T_b)
        & class_Int_Onumber(T_b) )
     => ( hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)) = V_a_2
      <=> ( ( hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2) != c_Groups_Ozero__class_Ozero(T_b)
           => V_b_2 = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)) )
          & ( hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2) = c_Groups_Ozero__class_Ozero(T_b)
           => V_a_2 = c_Groups_Ozero__class_Ozero(T_b) ) ) ) ) )).

fof(fact_divide__eq__eq__number__of,axiom,(
    ! [V_w_2,V_c_2,V_b_2,T_b] :
      ( ( class_Fields_Ofield__inverse__zero(T_b)
        & class_Int_Onumber(T_b) )
     => ( hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),V_c_2) = hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)
      <=> ( ( V_c_2 != c_Groups_Ozero__class_Ozero(T_b)
           => V_b_2 = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)),V_c_2) )
          & ( V_c_2 = c_Groups_Ozero__class_Ozero(T_b)
           => hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2) = c_Groups_Ozero__class_Ozero(T_b) ) ) ) ) )).

fof(fact_eq__divide__eq__number__of,axiom,(
    ! [V_c_2,V_b_2,V_w_2,T_b] :
      ( ( class_Fields_Ofield__inverse__zero(T_b)
        & class_Int_Onumber(T_b) )
     => ( hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),V_c_2)
      <=> ( ( V_c_2 != c_Groups_Ozero__class_Ozero(T_b)
           => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)),V_c_2) = V_b_2 )
          & ( V_c_2 = c_Groups_Ozero__class_Ozero(T_b)
           => hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2) = c_Groups_Ozero__class_Ozero(T_b) ) ) ) ) )).

fof(fact_eq__divide__eq__number__of1,axiom,(
    ! [V_w_2,V_b_2,V_a_2,T_b] :
      ( ( class_Fields_Ofield__inverse__zero(T_b)
        & class_Int_Onumber(T_b) )
     => ( V_a_2 = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2))
      <=> ( ( hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2) != c_Groups_Ozero__class_Ozero(T_b)
           => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)) = V_b_2 )
          & ( hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2) = c_Groups_Ozero__class_Ozero(T_b)
           => V_a_2 = c_Groups_Ozero__class_Ozero(T_b) ) ) ) ) )).

fof(fact_less__divide__eq__number__of1,axiom,(
    ! [V_w_2,V_b_2,V_a_2,T_b] :
      ( ( class_Fields_Olinordered__field__inverse__zero(T_b)
        & class_Int_Onumber(T_b) )
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2))))
      <=> ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)))
           => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2))),V_b_2)) )
          & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)))
           => ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)),c_Groups_Ozero__class_Ozero(T_b)))
               => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)))) )
              & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)),c_Groups_Ozero__class_Ozero(T_b)))
               => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),c_Groups_Ozero__class_Ozero(T_b))) ) ) ) ) ) ) )).

fof(fact_less__divide__eq__number__of,axiom,(
    ! [V_c_2,V_b_2,V_w_2,T_b] :
      ( ( class_Fields_Olinordered__field__inverse__zero(T_b)
        & class_Int_Onumber(T_b) )
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),V_c_2)))
      <=> ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_c_2))
           => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)),V_c_2)),V_b_2)) )
          & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_c_2))
           => ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),c_Groups_Ozero__class_Ozero(T_b)))
               => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)),V_c_2))) )
              & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),c_Groups_Ozero__class_Ozero(T_b)))
               => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)),c_Groups_Ozero__class_Ozero(T_b))) ) ) ) ) ) ) )).

fof(fact_divide__less__eq__number__of,axiom,(
    ! [V_w_2,V_c_2,V_b_2,T_b] :
      ( ( class_Fields_Olinordered__field__inverse__zero(T_b)
        & class_Int_Onumber(T_b) )
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),V_c_2)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)))
      <=> ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_c_2))
           => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)),V_c_2))) )
          & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_c_2))
           => ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),c_Groups_Ozero__class_Ozero(T_b)))
               => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)),V_c_2)),V_b_2)) )
              & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),c_Groups_Ozero__class_Ozero(T_b)))
               => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2))) ) ) ) ) ) ) )).

fof(fact_divide__less__eq__number__of1,axiom,(
    ! [V_a_2,V_w_2,V_b_2,T_b] :
      ( ( class_Fields_Olinordered__field__inverse__zero(T_b)
        & class_Int_Onumber(T_b) )
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2))),V_a_2))
      <=> ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)))
           => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)))) )
          & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)))
           => ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)),c_Groups_Ozero__class_Ozero(T_b)))
               => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2))),V_b_2)) )
              & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)),c_Groups_Ozero__class_Ozero(T_b)))
               => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_a_2)) ) ) ) ) ) ) )).

fof(fact_le__divide__eq__number__of1,axiom,(
    ! [V_w_2,V_b_2,V_a_2,T_b] :
      ( ( class_Fields_Olinordered__field__inverse__zero(T_b)
        & class_Int_Onumber(T_b) )
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2))))
      <=> ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)))
           => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2))),V_b_2)) )
          & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)))
           => ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)),c_Groups_Ozero__class_Ozero(T_b)))
               => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)))) )
              & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)),c_Groups_Ozero__class_Ozero(T_b)))
               => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),c_Groups_Ozero__class_Ozero(T_b))) ) ) ) ) ) ) )).

fof(fact_le__divide__eq__number__of,axiom,(
    ! [V_c_2,V_b_2,V_w_2,T_b] :
      ( ( class_Fields_Olinordered__field__inverse__zero(T_b)
        & class_Int_Onumber(T_b) )
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),V_c_2)))
      <=> ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_c_2))
           => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)),V_c_2)),V_b_2)) )
          & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_c_2))
           => ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),c_Groups_Ozero__class_Ozero(T_b)))
               => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)),V_c_2))) )
              & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),c_Groups_Ozero__class_Ozero(T_b)))
               => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)),c_Groups_Ozero__class_Ozero(T_b))) ) ) ) ) ) ) )).

fof(fact_divide__le__eq__number__of,axiom,(
    ! [V_w_2,V_c_2,V_b_2,T_b] :
      ( ( class_Fields_Olinordered__field__inverse__zero(T_b)
        & class_Int_Onumber(T_b) )
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),V_c_2)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)))
      <=> ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_c_2))
           => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)),V_c_2))) )
          & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_c_2))
           => ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),c_Groups_Ozero__class_Ozero(T_b)))
               => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)),V_c_2)),V_b_2)) )
              & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),c_Groups_Ozero__class_Ozero(T_b)))
               => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2))) ) ) ) ) ) ) )).

fof(fact_divide__le__eq__number__of1,axiom,(
    ! [V_a_2,V_w_2,V_b_2,T_b] :
      ( ( class_Fields_Olinordered__field__inverse__zero(T_b)
        & class_Int_Onumber(T_b) )
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2))),V_a_2))
      <=> ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)))
           => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)))) )
          & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)))
           => ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)),c_Groups_Ozero__class_Ozero(T_b)))
               => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2))),V_b_2)) )
              & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)),c_Groups_Ozero__class_Ozero(T_b)))
               => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_a_2)) ) ) ) ) ) ) )).

fof(fact_setprod__pos,axiom,(
    ! [V_f_2,V_A_2,T_b,T_c] :
      ( class_Rings_Olinordered__semidom(T_c)
     => ( ! [B_x] :
            ( hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2))
           => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(V_f_2,B_x))) )
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_f_2),V_A_2))) ) ) )).

fof(fact_setprod__nonneg,axiom,(
    ! [V_f_2,V_A_2,T_b,T_c] :
      ( class_Rings_Olinordered__semidom(T_c)
     => ( ! [B_x] :
            ( hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2))
           => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(V_f_2,B_x))) )
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_f_2),V_A_2))) ) ) )).

fof(fact_number__of__succ,axiom,(
    ! [V_k,T_a] :
      ( class_Int_Onumber__ring(T_a)
     => hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_Osucc(V_k)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Groups_Oone__class_Oone(T_a)),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_k)) ) )).

fof(fact_com_Osize_I6_J,axiom,(
    ! [V_com_2,V_fun_2] : c_Com_Ocom_Ocom__size(c_Com_Ocom_OWhile(V_fun_2,V_com_2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Com_Ocom_Ocom__size(V_com_2)),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) )).

fof(fact_com_Osize_I14_J,axiom,(
    ! [V_com_2,V_fun_2] : hAPP(c_Nat_Osize__class_Osize(tc_Com_Ocom),c_Com_Ocom_OWhile(V_fun_2,V_com_2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_Com_Ocom),V_com_2)),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) )).

fof(fact_field__le__epsilon,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Fields_Olinordered__field(T_a)
     => ( ! [B_e] :
            ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),B_e))
           => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_y),B_e))) )
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y)) ) ) )).

fof(fact_pair__box_Osize_I1_J,axiom,(
    ! [V_b_2,V_a_2,V_fb_2,V_fa_2,T_c,T_b] : c_Nitpick_Opair__box_Opair__box__size(T_b,T_c,V_fa_2,V_fb_2,c_Nitpick_Opair__box_OPairBox(T_b,T_c,V_a_2,V_b_2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(V_fa_2,V_a_2)),hAPP(V_fb_2,V_b_2))),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) )).

fof(fact_nat_Osize_I2_J,axiom,(
    ! [V_nat] : c_Nat_Onat_Onat__size(hAPP(c_Nat_OSuc,V_nat)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Nat_Onat_Onat__size(V_nat)),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) )).

fof(fact_less__eq__number__of__int__code,axiom,(
    ! [V_l_2,V_k_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_k_2)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_l_2)))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k_2),V_l_2)) ) )).

fof(fact_less__number__of__int__code,axiom,(
    ! [V_l_2,V_k_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_k_2)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_l_2)))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),V_l_2)) ) )).

fof(fact_plus__numeral__code_I9_J,axiom,(
    ! [V_w,V_v] : hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_v),V_w)) )).

fof(fact_times__numeral__code_I5_J,axiom,(
    ! [V_w,V_v] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_v),V_w)) )).

fof(fact_less__bin__lemma,axiom,(
    ! [V_l_2,V_k_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),V_l_2))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_k_2),V_l_2)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) ) )).

fof(fact_zmult__zless__mono2,axiom,(
    ! [V_k,V_j,V_i] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_i),V_j))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_k))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_k),V_i)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_k),V_j))) ) ) )).

fof(fact_zadd__0,axiom,(
    ! [V_z] : hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z) = V_z )).

fof(fact_zadd__0__right,axiom,(
    ! [V_z] : hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_z),c_Groups_Ozero__class_Ozero(tc_Int_Oint)) = V_z )).

fof(fact_odd__nonzero,axiom,(
    ! [V_z] : hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),V_z)),V_z) != c_Groups_Ozero__class_Ozero(tc_Int_Oint) )).

fof(fact_le__imp__0__less,axiom,(
    ! [V_z] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),V_z))) ) )).

fof(fact_int__one__le__iff__zero__less,axiom,(
    ! [V_z_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),V_z_2))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z_2)) ) )).

fof(fact_odd__less__0,axiom,(
    ! [V_z_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),V_z_2)),V_z_2)),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_z_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) ) )).

fof(fact_int__0__less__1,axiom,(
    hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Groups_Oone__class_Oone(tc_Int_Oint))) )).

fof(fact_pos__zmult__eq__1__iff,axiom,(
    ! [V_n_2,V_m_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_m_2))
     => ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_m_2),V_n_2) = c_Groups_Oone__class_Oone(tc_Int_Oint)
      <=> ( V_m_2 = c_Groups_Oone__class_Oone(tc_Int_Oint)
          & V_n_2 = c_Groups_Oone__class_Oone(tc_Int_Oint) ) ) ) )).

fof(fact_zle__diff1__eq,axiom,(
    ! [V_z_2,V_w_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_w_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_z_2),c_Groups_Oone__class_Oone(tc_Int_Oint))))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_w_2),V_z_2)) ) )).

fof(fact_zless__add1__eq,axiom,(
    ! [V_z_2,V_w_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_w_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_z_2),c_Groups_Oone__class_Oone(tc_Int_Oint))))
    <=> ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_w_2),V_z_2))
        | V_w_2 = V_z_2 ) ) )).

fof(fact_zle__add1__eq__le,axiom,(
    ! [V_z_2,V_w_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_w_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_z_2),c_Groups_Oone__class_Oone(tc_Int_Oint))))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_w_2),V_z_2)) ) )).

fof(fact_add1__zle__eq,axiom,(
    ! [V_z_2,V_w_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_w_2),c_Groups_Oone__class_Oone(tc_Int_Oint))),V_z_2))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_w_2),V_z_2)) ) )).

fof(fact_zless__imp__add1__zle,axiom,(
    ! [V_z,V_w] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_w),V_z))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_w),c_Groups_Oone__class_Oone(tc_Int_Oint))),V_z)) ) )).

fof(fact_zadd__zless__mono,axiom,(
    ! [V_z,V_z_H,V_w,V_w_H] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_w_H),V_w))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_z_H),V_z))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_w_H),V_z_H)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_w),V_z))) ) ) )).

fof(fact_succ__def,axiom,(
    ! [V_k] : c_Int_Osucc(V_k) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_k),c_Groups_Oone__class_Oone(tc_Int_Oint)) )).

fof(fact_zadd__left__mono,axiom,(
    ! [V_k,V_j,V_i] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_i),V_j))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_k),V_i)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_k),V_j))) ) )).

fof(fact_zmult__1,axiom,(
    ! [V_z] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),V_z) = V_z )).

fof(fact_zmult__1__right,axiom,(
    ! [V_z] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_z),c_Groups_Oone__class_Oone(tc_Int_Oint)) = V_z )).

fof(fact_zmult__assoc,axiom,(
    ! [V_z3,V_z2,V_z1] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_z1),V_z2)),V_z3) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_z1),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_z2),V_z3)) )).

fof(fact_zmult__commute,axiom,(
    ! [V_w,V_z] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_z),V_w) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_w),V_z) )).

fof(fact_zadd__assoc,axiom,(
    ! [V_z3,V_z2,V_z1] : hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_z1),V_z2)),V_z3) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_z1),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_z2),V_z3)) )).

fof(fact_zadd__left__commute,axiom,(
    ! [V_z,V_y,V_x] : hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_x),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_y),V_z)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_y),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_x),V_z)) )).

fof(fact_zadd__commute,axiom,(
    ! [V_w,V_z] : hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_z),V_w) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_w),V_z) )).

fof(fact_zadd__zmult__distrib2,axiom,(
    ! [V_z2,V_z1,V_w] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_w),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_z1),V_z2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_w),V_z1)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_w),V_z2)) )).

fof(fact_zadd__zmult__distrib,axiom,(
    ! [V_w,V_z2,V_z1] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_z1),V_z2)),V_w) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_z1),V_w)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_z2),V_w)) )).

fof(fact_zdiff__zmult__distrib,axiom,(
    ! [V_w,V_z2,V_z1] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_z1),V_z2)),V_w) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_z1),V_w)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_z2),V_w)) )).

fof(fact_zdiff__zmult__distrib2,axiom,(
    ! [V_z2,V_z1,V_w] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_w),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_z1),V_z2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_w),V_z1)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_w),V_z2)) )).

fof(fact_zadd__strict__right__mono,axiom,(
    ! [V_k,V_j,V_i] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_i),V_j))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_i),V_k)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_j),V_k))) ) )).

fof(fact_zless__linear,axiom,(
    ! [V_y,V_x] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_x),V_y))
      | V_x = V_y
      | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_y),V_x)) ) )).

fof(fact_zle__refl,axiom,(
    ! [V_w] : hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_w),V_w)) )).

fof(fact_zle__linear,axiom,(
    ! [V_w,V_z] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_z),V_w))
      | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_w),V_z)) ) )).

fof(fact_zle__trans,axiom,(
    ! [V_k,V_j,V_i] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_i),V_j))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_j),V_k))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_i),V_k)) ) ) )).

fof(fact_zle__antisym,axiom,(
    ! [V_w,V_z] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_z),V_w))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_w),V_z))
       => V_z = V_w ) ) )).

fof(fact_zless__le,axiom,(
    ! [V_w_2,V_z_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_z_2),V_w_2))
    <=> ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_z_2),V_w_2))
        & V_z_2 != V_w_2 ) ) )).

fof(fact_com_Osimps_I5_J,axiom,(
    ! [V_com_H_2,V_fun_H_2,V_com_2,V_fun_2] :
      ( c_Com_Ocom_OWhile(V_fun_2,V_com_2) = c_Com_Ocom_OWhile(V_fun_H_2,V_com_H_2)
    <=> ( V_fun_2 = V_fun_H_2
        & V_com_2 = V_com_H_2 ) ) )).

fof(fact_pair__box_Oinject,axiom,(
    ! [V_b_H_2,V_a_H_2,V_b_2,V_a_2,T_c,T_b] :
      ( c_Nitpick_Opair__box_OPairBox(T_b,T_c,V_a_2,V_b_2) = c_Nitpick_Opair__box_OPairBox(T_b,T_c,V_a_H_2,V_b_H_2)
    <=> ( V_a_2 = V_a_H_2
        & V_b_2 = V_b_H_2 ) ) )).

fof(fact_nat_Osize_I1_J,axiom,(
    c_Nat_Onat_Onat__size(c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )).

fof(fact_com_Osimps_I52_J,axiom,(
    ! [V_com_H_2,V_fun_H_2,V_com2_2,V_com1_2,V_fun_2] : c_Com_Ocom_OCond(V_fun_2,V_com1_2,V_com2_2) != c_Com_Ocom_OWhile(V_fun_H_2,V_com_H_2) )).

fof(fact_com_Osimps_I53_J,axiom,(
    ! [V_com2_2,V_com1_2,V_fun_2,V_com_H_2,V_fun_H_2] : c_Com_Ocom_OWhile(V_fun_H_2,V_com_H_2) != c_Com_Ocom_OCond(V_fun_2,V_com1_2,V_com2_2) )).

fof(fact_com_Osimps_I46_J,axiom,(
    ! [V_com_H_2,V_fun_H_2,V_com2_2,V_com1_2] : c_Com_Ocom_OSemi(V_com1_2,V_com2_2) != c_Com_Ocom_OWhile(V_fun_H_2,V_com_H_2) )).

fof(fact_com_Osimps_I47_J,axiom,(
    ! [V_com2_2,V_com1_2,V_com_H_2,V_fun_H_2] : c_Com_Ocom_OWhile(V_fun_H_2,V_com_H_2) != c_Com_Ocom_OSemi(V_com1_2,V_com2_2) )).

fof(fact_com_Osimps_I38_J,axiom,(
    ! [V_com_H_2,V_fun_H_2,V_com_2,V_fun_2,V_loc_2] : c_Com_Ocom_OLocal(V_loc_2,V_fun_2,V_com_2) != c_Com_Ocom_OWhile(V_fun_H_2,V_com_H_2) )).

fof(fact_com_Osimps_I39_J,axiom,(
    ! [V_com_2,V_fun_2,V_loc_2,V_com_H_2,V_fun_H_2] : c_Com_Ocom_OWhile(V_fun_H_2,V_com_H_2) != c_Com_Ocom_OLocal(V_loc_2,V_fun_2,V_com_2) )).

fof(fact_com_Osimps_I16_J,axiom,(
    ! [V_com_H_2,V_fun_H_2] : c_Com_Ocom_OSKIP != c_Com_Ocom_OWhile(V_fun_H_2,V_com_H_2) )).

fof(fact_com_Osimps_I17_J,axiom,(
    ! [V_com_H_2,V_fun_H_2] : c_Com_Ocom_OWhile(V_fun_H_2,V_com_H_2) != c_Com_Ocom_OSKIP )).

fof(fact_self__quotient__aux2,axiom,(
    ! [V_q,V_r,V_a] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a))
     => ( V_a = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_r),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),V_q))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_r))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_q),c_Groups_Oone__class_Oone(tc_Int_Oint))) ) ) ) )).

fof(fact_self__quotient__aux1,axiom,(
    ! [V_q,V_r,V_a] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a))
     => ( V_a = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_r),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),V_q))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_r),V_a))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),V_q)) ) ) ) )).

fof(fact_q__pos__lemma,axiom,(
    ! [V_r_H,V_q_H,V_b_H] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b_H),V_q_H)),V_r_H)))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_r_H),V_b_H))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b_H))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_q_H)) ) ) ) )).

fof(fact_q__neg__lemma,axiom,(
    ! [V_r_H,V_q_H,V_b_H] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b_H),V_q_H)),V_r_H)),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_r_H))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b_H))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_q_H),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) ) ) ) )).

fof(fact_unique__quotient__lemma,axiom,(
    ! [V_r,V_q,V_r_H,V_q_H,V_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),V_q_H)),V_r_H)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),V_q)),V_r)))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_r_H))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_r_H),V_b))
         => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_r),V_b))
           => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_q_H),V_q)) ) ) ) ) )).

fof(fact_zdiv__mono2__lemma,axiom,(
    ! [V_r_H,V_q_H,V_b_H,V_r,V_q,V_b] :
      ( hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),V_q)),V_r) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b_H),V_q_H)),V_r_H)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b_H),V_q_H)),V_r_H)))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_r_H),V_b_H))
         => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_r))
           => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b_H))
             => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_b_H),V_b))
               => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_q),V_q_H)) ) ) ) ) ) ) )).

fof(fact_unique__quotient__lemma__neg,axiom,(
    ! [V_r,V_q,V_r_H,V_q_H,V_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),V_q_H)),V_r_H)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),V_q)),V_r)))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_r),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b),V_r))
         => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b),V_r_H))
           => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_q),V_q_H)) ) ) ) ) )).

fof(fact_number__of__is__id,axiom,(
    ! [V_k] : hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_k) = V_k )).

fof(fact_int__0__neq__1,axiom,(
    c_Groups_Ozero__class_Ozero(tc_Int_Oint) != c_Groups_Oone__class_Oone(tc_Int_Oint) )).

fof(fact_zdiv__mono2__neg__lemma,axiom,(
    ! [V_r_H,V_q_H,V_b_H,V_r,V_q,V_b] :
      ( hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),V_q)),V_r) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b_H),V_q_H)),V_r_H)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b_H),V_q_H)),V_r_H)),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_r),V_b))
         => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_r_H))
           => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b_H))
             => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_b_H),V_b))
               => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_q_H),V_q)) ) ) ) ) ) ) )).

fof(fact_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J,axiom,(
    ! [V_y,V_x] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_y))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_x),V_y))) ) ) )).

fof(fact_Nat__Transfer_Otransfer__nat__int__function__closures_I6_J,axiom,(
    hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Groups_Oone__class_Oone(tc_Int_Oint))) )).

fof(fact_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J,axiom,(
    ! [V_y,V_x] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_y))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_x),V_y))) ) ) )).

fof(fact_pair__box_Osize_I2_J,axiom,(
    ! [V_b,V_a,T_b,T_a] : hAPP(c_Nat_Osize__class_Osize(tc_Nitpick_Opair__box(T_a,T_b)),c_Nitpick_Opair__box_OPairBox(T_a,T_b,V_a,V_b)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )).

fof(fact_pair__box_Orecs,axiom,(
    ! [V_b_2,V_a_2,V_f1_2,T_b,T_d,T_c] : c_Nitpick_Opair__box_Opair__box__rec(T_c,T_d,T_b,V_f1_2,c_Nitpick_Opair__box_OPairBox(T_c,T_d,V_a_2,V_b_2)) = hAPP(hAPP(V_f1_2,V_a_2),V_b_2) )).

fof(fact_pair__box_Osimps_I2_J,axiom,(
    ! [V_b_2,V_a_2,V_f1_2,T_b,T_d,T_c] : c_Nitpick_Opair__box_Opair__box__case(T_c,T_d,T_b,V_f1_2,c_Nitpick_Opair__box_OPairBox(T_c,T_d,V_a_2,V_b_2)) = hAPP(hAPP(V_f1_2,V_a_2),V_b_2) )).

fof(fact_nat_Osize_I4_J,axiom,(
    ! [V_nat] : hAPP(c_Nat_Osize__class_Osize(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_nat)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_Nat_Onat),V_nat)),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) )).

fof(fact_nat__size,axiom,(
    ! [V_n] : hAPP(c_Nat_Osize__class_Osize(tc_Nat_Onat),V_n) = V_n )).

fof(fact_nat_Osize_I3_J,axiom,(
    hAPP(c_Nat_Osize__class_Osize(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )).

fof(fact_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,(
    hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) )).

fof(fact_Suc__nat__number__of__add,axiom,(
    ! [V_n,V_v] :
      ( ( hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)))
       => hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),V_n)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),V_n) )
      & ( ~ hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)))
       => hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),V_n)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_Osucc(V_v))),V_n) ) ) )).

fof(fact_tsub__def,axiom,(
    ! [V_x,V_y] :
      ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_y),V_x))
       => c_Nat__Transfer_Otsub(V_x,V_y) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_x),V_y) )
      & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_y),V_x))
       => c_Nat__Transfer_Otsub(V_x,V_y) = c_Groups_Ozero__class_Ozero(tc_Int_Oint) ) ) )).

fof(fact_Suc__nat__number__of,axiom,(
    ! [V_v] :
      ( ( hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)))
       => hAPP(c_Nat_OSuc,hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)) = c_Groups_Oone__class_Oone(tc_Nat_Onat) )
      & ( ~ hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)))
       => hAPP(c_Nat_OSuc,hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_Osucc(V_v)) ) ) )).

fof(fact_of__nat__number__of__lemma,axiom,(
    ! [V_v,T_a] :
      ( class_Int_Onumber__ring(T_a)
     => ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)))
         => hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_v) )
        & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)))
         => hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)) = c_Groups_Ozero__class_Ozero(T_a) ) ) ) )).

fof(fact_nat__number__of__add__1,axiom,(
    ! [V_v] :
      ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_v),c_Int_OPls))
       => hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = c_Groups_Oone__class_Oone(tc_Nat_Onat) )
      & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_v),c_Int_OPls))
       => hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_Osucc(V_v)) ) ) )).

fof(fact_nat__1__add__number__of,axiom,(
    ! [V_v] :
      ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_v),c_Int_OPls))
       => hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)) = c_Groups_Oone__class_Oone(tc_Nat_Onat) )
      & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_v),c_Int_OPls))
       => hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_Osucc(V_v)) ) ) )).

fof(fact_mult__nat__number__of,axiom,(
    ! [V_v_H,V_v] :
      ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_v),c_Int_OPls))
       => hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )
      & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_v),c_Int_OPls))
       => hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H)) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_v),V_v_H)) ) ) )).

fof(fact_Nat__Transfer_Otransfer__int__nat__functions_I3_J,axiom,(
    ! [V_y,V_x] : c_Nat__Transfer_Otsub(hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_x),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_y)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_x),V_y)) )).

fof(fact_not__neg__int,axiom,(
    ! [V_n] : ~ hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n))) )).

fof(fact_of__nat__eq__iff,axiom,(
    ! [V_n_2,V_m_2,T_b] :
      ( class_Nat_Osemiring__char__0(T_b)
     => ( hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_b),V_m_2) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_b),V_n_2)
      <=> V_m_2 = V_n_2 ) ) )).

fof(fact_not__neg__number__of__Pls,axiom,(
    ~ hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OPls))) )).

fof(fact_Pls__def,axiom,(
    c_Int_OPls = c_Groups_Ozero__class_Ozero(tc_Int_Oint) )).

fof(fact_rel__simps_I19_J,axiom,(
    hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OPls),c_Int_OPls)) )).

fof(fact_rel__simps_I2_J,axiom,(
    ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OPls),c_Int_OPls)) )).

fof(fact_add__Pls,axiom,(
    ! [V_k] : hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Int_OPls),V_k) = V_k )).

fof(fact_add__Pls__right,axiom,(
    ! [V_k] : hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_k),c_Int_OPls) = V_k )).

fof(fact_mult__Pls,axiom,(
    ! [V_w] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Int_OPls),V_w) = c_Int_OPls )).

fof(fact_diff__bin__simps_I1_J,axiom,(
    ! [V_k] : hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_k),c_Int_OPls) = V_k )).

fof(fact_not__neg__0,axiom,(
    ~ hBOOL(hAPP(c_Nat__Numeral_Oneg,c_Groups_Ozero__class_Ozero(tc_Int_Oint))) )).

fof(fact_not__neg__1,axiom,(
    ~ hBOOL(hAPP(c_Nat__Numeral_Oneg,c_Groups_Oone__class_Oone(tc_Int_Oint))) )).

fof(fact_int__nat__number__of,axiom,(
    ! [V_v] :
      ( ( hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)))
       => hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)) = c_Groups_Ozero__class_Ozero(tc_Int_Oint) )
      & ( ~ hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)))
       => hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v) ) ) )).

fof(fact_int__0,axiom,(
    hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = c_Groups_Ozero__class_Ozero(tc_Int_Oint) )).

fof(fact_int__eq__0__conv,axiom,(
    ! [V_n_2] :
      ( hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n_2) = c_Groups_Ozero__class_Ozero(tc_Int_Oint)
    <=> V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )).

fof(fact_transfer__int__nat__numerals_I1_J,axiom,(
    c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) )).

fof(fact_zero__zle__int,axiom,(
    ! [V_n] : hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n))) )).

fof(fact_transfer__int__nat__quantifiers_I1_J,axiom,(
    ! [V_P_2] :
      ( ! [B_x] :
          ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B_x))
         => hBOOL(hAPP(V_P_2,B_x)) )
    <=> ! [B_x] : hBOOL(hAPP(V_P_2,hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),B_x))) ) )).

fof(fact_transfer__int__nat__quantifiers_I2_J,axiom,(
    ! [V_P_2] :
      ( ? [B_x] :
          ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B_x))
          & hBOOL(hAPP(V_P_2,B_x)) )
    <=> ? [B_x] : hBOOL(hAPP(V_P_2,hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),B_x))) ) )).

fof(fact_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J,axiom,(
    ! [V_z] : hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_z))) )).

fof(fact_int__less__0__conv,axiom,(
    ! [V_k] : ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_k)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) )).

fof(fact_zless__int,axiom,(
    ! [V_n_2,V_m_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m_2)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n_2)))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) ) )).

fof(fact_transfer__int__nat__relations_I2_J,axiom,(
    ! [V_y_2,V_x_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_x_2)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_y_2)))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_x_2),V_y_2)) ) )).

fof(fact_zle__int,axiom,(
    ! [V_n_2,V_m_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m_2)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n_2)))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2)) ) )).

fof(fact_transfer__int__nat__relations_I3_J,axiom,(
    ! [V_y_2,V_x_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_x_2)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_y_2)))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_x_2),V_y_2)) ) )).

fof(fact_zadd__int__left,axiom,(
    ! [V_z,V_n,V_m] : hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n)),V_z)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n))),V_z) )).

fof(fact_zadd__int,axiom,(
    ! [V_n,V_m] : hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n)) )).

fof(fact_Nat__Transfer_Otransfer__int__nat__functions_I1_J,axiom,(
    ! [V_y,V_x] : hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_x)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_y)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_x),V_y)) )).

fof(fact_zle__iff__zadd,axiom,(
    ! [V_z_2,V_w_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_w_2),V_z_2))
    <=> ? [B_n] : V_z_2 = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_w_2),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),B_n)) ) )).

fof(fact_zero__le__imp__of__nat,axiom,(
    ! [V_m,T_a] :
      ( class_Rings_Olinordered__semidom(T_a)
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),V_m))) ) )).

fof(fact_of__nat__0__le__iff,axiom,(
    ! [V_n,T_a] :
      ( class_Rings_Olinordered__semidom(T_a)
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),V_n))) ) )).

fof(fact_of__nat__less__0__iff,axiom,(
    ! [V_m,T_a] :
      ( class_Rings_Olinordered__semidom(T_a)
     => ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),V_m)),c_Groups_Ozero__class_Ozero(T_a))) ) )).

fof(fact_number__of__Pls,axiom,(
    ! [T_a] :
      ( class_Int_Onumber__ring(T_a)
     => hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OPls) = c_Groups_Ozero__class_Ozero(T_a) ) )).

fof(fact_semiring__norm_I112_J,axiom,(
    ! [T_a] :
      ( class_Int_Onumber__ring(T_a)
     => c_Groups_Ozero__class_Ozero(T_a) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OPls) ) )).

fof(fact_of__nat__number__of__eq,axiom,(
    ! [V_v,T_a] :
      ( class_Int_Onumber__ring(T_a)
     => ( ( hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)))
         => hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)) = c_Groups_Ozero__class_Ozero(T_a) )
        & ( ~ hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)))
         => hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_v) ) ) ) )).

fof(fact_int__mult,axiom,(
    ! [V_n,V_m] : hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n)) )).

fof(fact_zmult__int,axiom,(
    ! [V_n,V_m] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n)) )).

fof(fact_Nat__Transfer_Otransfer__int__nat__functions_I2_J,axiom,(
    ! [V_y,V_x] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_x)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_y)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_x),V_y)) )).

fof(fact_add__numeral__0__right,axiom,(
    ! [V_a,T_a] :
      ( class_Int_Onumber__ring(T_a)
     => hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OPls)) = V_a ) )).

fof(fact_add__numeral__0,axiom,(
    ! [V_a,T_a] :
      ( class_Int_Onumber__ring(T_a)
     => hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OPls)),V_a) = V_a ) )).

fof(fact_of__nat__0,axiom,(
    ! [T_a] :
      ( class_Rings_Osemiring__1(T_a)
     => hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = c_Groups_Ozero__class_Ozero(T_a) ) )).

fof(fact_of__nat__less__imp__less,axiom,(
    ! [V_n,V_m,T_a] :
      ( class_Rings_Olinordered__semidom(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),V_m)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),V_n)))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) ) ) )).

fof(fact_less__imp__of__nat__less,axiom,(
    ! [V_n,V_m,T_a] :
      ( class_Rings_Olinordered__semidom(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),V_m)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),V_n))) ) ) )).

fof(fact_of__nat__less__iff,axiom,(
    ! [V_n_2,V_m_2,T_b] :
      ( class_Rings_Olinordered__semidom(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_b),V_m_2)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_b),V_n_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) ) ) )).

fof(fact_int__1,axiom,(
    hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = c_Groups_Oone__class_Oone(tc_Int_Oint) )).

fof(fact_transfer__int__nat__numerals_I2_J,axiom,(
    c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Nat_Onat)) )).

fof(fact_of__nat__le__iff,axiom,(
    ! [V_n_2,V_m_2,T_b] :
      ( class_Rings_Olinordered__semidom(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_b),V_m_2)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_b),V_n_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2)) ) ) )).

fof(fact_of__nat__add,axiom,(
    ! [V_n,V_m,T_a] :
      ( class_Rings_Osemiring__1(T_a)
     => hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),V_m)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),V_n)) ) )).

fof(fact_bin__less__0__simps_I1_J,axiom,(
    ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OPls),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) )).

fof(fact_nat__number__of__Pls,axiom,(
    hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OPls) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )).

fof(fact_semiring__norm_I113_J,axiom,(
    c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OPls) )).

fof(fact_of__nat__mult,axiom,(
    ! [V_n,V_m,T_a] :
      ( class_Rings_Osemiring__1(T_a)
     => hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),V_m)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),V_n)) ) )).

fof(fact_zero__is__num__zero,axiom,(
    c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OPls) )).

fof(fact_of__nat__1,axiom,(
    ! [T_a] :
      ( class_Rings_Osemiring__1(T_a)
     => hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = c_Groups_Oone__class_Oone(T_a) ) )).

fof(fact_Ints__of__nat,axiom,(
    ! [V_n_2,T_b] :
      ( class_Rings_Oring__1(T_b)
     => hBOOL(hAPP(hAPP(c_member(T_b),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_b),V_n_2)),c_Int_Oring__1__class_OInts(T_b))) ) )).

fof(fact_not__neg__eq__ge__0,axiom,(
    ! [V_x_2] :
      ( ~ hBOOL(hAPP(c_Nat__Numeral_Oneg,V_x_2))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x_2)) ) )).

fof(fact_neg__def,axiom,(
    ! [V_Z_2] :
      ( hBOOL(hAPP(c_Nat__Numeral_Oneg,V_Z_2))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_Z_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) ) )).

fof(fact_int__le__0__conv,axiom,(
    ! [V_n_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n_2)),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
    <=> V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )).

fof(fact_int__Suc0__eq__1,axiom,(
    hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) = c_Groups_Oone__class_Oone(tc_Int_Oint) )).

fof(fact_zless__iff__Suc__zadd,axiom,(
    ! [V_z_2,V_w_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_w_2),V_z_2))
    <=> ? [B_n] : V_z_2 = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_w_2),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Nat_OSuc,B_n))) ) )).

fof(fact_int__Suc,axiom,(
    ! [V_m] : hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Nat_OSuc,V_m)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m)) )).

fof(fact_zdiff__int,axiom,(
    ! [V_m,V_n] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n),V_m))
     => hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)) ) )).

fof(fact_of__nat__Suc,axiom,(
    ! [V_m,T_a] :
      ( class_Rings_Osemiring__1(T_a)
     => hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),hAPP(c_Nat_OSuc,V_m)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Groups_Oone__class_Oone(T_a)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),V_m)) ) )).

fof(fact_divide__Numeral0,axiom,(
    ! [V_x,T_a] :
      ( ( class_Fields_Ofield__inverse__zero(T_a)
        & class_Int_Onumber__ring(T_a) )
     => hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OPls)) = c_Groups_Ozero__class_Ozero(T_a) ) )).

fof(fact_eq__0__number__of,axiom,(
    ! [V_v_2] :
      ( c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2)
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_v_2),c_Int_OPls)) ) )).

fof(fact_eq__number__of__0,axiom,(
    ! [V_v_2] :
      ( hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_v_2),c_Int_OPls)) ) )).

fof(fact_less__nat__number__of,axiom,(
    ! [V_v_H_2,V_v_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H_2)))
    <=> ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_v_2),V_v_H_2))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OPls),V_v_H_2)) )
        & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_v_2),V_v_H_2)) ) ) )).

fof(fact_of__nat__diff,axiom,(
    ! [V_m,V_n,T_a] :
      ( class_Rings_Oring__1(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n),V_m))
       => hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),V_m)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),V_n)) ) ) )).

fof(fact_le__nat__number__of,axiom,(
    ! [V_v_H_2,V_v_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H_2)))
    <=> ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_v_2),V_v_H_2))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_v_2),c_Int_OPls)) ) ) )).

fof(fact_neg__imp__number__of__eq__0,axiom,(
    ! [V_v] :
      ( hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)))
     => hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )).

fof(fact_zero__less__int__conv,axiom,(
    ! [V_n_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n_2)))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n_2)) ) )).

fof(fact_zmult__zless__mono2__lemma,axiom,(
    ! [V_k,V_j,V_i] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_i),V_j))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_k))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_k)),V_i)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_k)),V_j))) ) ) )).

fof(fact_le__special_I1_J,axiom,(
    ! [V_y_2,T_b] :
      ( ( class_Int_Onumber__ring(T_b)
        & class_Rings_Olinordered__idom(T_b) )
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_y_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OPls),V_y_2)) ) ) )).

fof(fact_le__special_I3_J,axiom,(
    ! [V_x_2,T_b] :
      ( ( class_Int_Onumber__ring(T_b)
        & class_Rings_Olinordered__idom(T_b) )
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_x_2)),c_Groups_Ozero__class_Ozero(T_b)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_x_2),c_Int_OPls)) ) ) )).

fof(fact_less__special_I1_J,axiom,(
    ! [V_y_2,T_b] :
      ( ( class_Int_Onumber__ring(T_b)
        & class_Rings_Olinordered__idom(T_b) )
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_y_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OPls),V_y_2)) ) ) )).

fof(fact_less__special_I3_J,axiom,(
    ! [V_x_2,T_b] :
      ( ( class_Int_Onumber__ring(T_b)
        & class_Rings_Olinordered__idom(T_b) )
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_x_2)),c_Groups_Ozero__class_Ozero(T_b)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_x_2),c_Int_OPls)) ) ) )).

fof(fact_of__nat__0__less__iff,axiom,(
    ! [V_n_2,T_b] :
      ( class_Rings_Olinordered__semidom(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_b),V_n_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n_2)) ) ) )).

fof(fact_less__0__number__of,axiom,(
    ! [V_v_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2)))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OPls),V_v_2)) ) )).

fof(fact_add__nat__number__of,axiom,(
    ! [V_v_H,V_v] :
      ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_v),c_Int_OPls))
       => hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H)) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H) )
      & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_v),c_Int_OPls))
       => ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_v_H),c_Int_OPls))
           => hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H)) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v) )
          & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_v_H),c_Int_OPls))
           => hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H)) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_v),V_v_H)) ) ) ) ) )).

fof(fact_eq__nat__number__of,axiom,(
    ! [V_v_H_2,V_v_2] :
      ( hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H_2)
    <=> ( ( hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_2)))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_H_2)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) )
        & ( ~ hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_2)))
         => ( ( hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_H_2)))
             => hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_2) = c_Groups_Ozero__class_Ozero(tc_Int_Oint) )
            & ( ~ hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_H_2)))
             => V_v_2 = V_v_H_2 ) ) ) ) ) )).

fof(fact_nat__number__of__add__left,axiom,(
    ! [V_k,V_v_H,V_v] :
      ( ( hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)))
       => hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H)),V_k)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H)),V_k) )
      & ( ~ hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)))
       => ( ( hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_H)))
           => hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H)),V_k)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),V_k) )
          & ( ~ hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_H)))
           => hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H)),V_k)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_v),V_v_H))),V_k) ) ) ) ) )).

fof(fact_Nat__Transfer_Otransfer__nat__int__function__closures_I3_J,axiom,(
    ! [V_y,V_x] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_y))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Nat__Transfer_Otsub(V_x,V_y))) ) ) )).

fof(fact_tsub__eq,axiom,(
    ! [V_x,V_y] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_y),V_x))
     => c_Nat__Transfer_Otsub(V_x,V_y) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_x),V_y) ) )).

fof(fact_nat__number__of__mult__left,axiom,(
    ! [V_k,V_v_H,V_v] :
      ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_v),c_Int_OPls))
       => hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H)),V_k)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )
      & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_v),c_Int_OPls))
       => hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H)),V_k)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_v),V_v_H))),V_k) ) ) )).

fof(fact_zdiff__int__split,axiom,(
    ! [V_y_2,V_x_2,V_P_2] :
      ( hBOOL(hAPP(V_P_2,hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_x_2),V_y_2))))
    <=> ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_y_2),V_x_2))
         => hBOOL(hAPP(V_P_2,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_x_2)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_y_2)))) )
        & ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_x_2),V_y_2))
         => hBOOL(hAPP(V_P_2,c_Groups_Ozero__class_Ozero(tc_Int_Oint))) ) ) ) )).

fof(fact_number__of2,axiom,(
    hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OPls))) )).

fof(fact_nat__number__of__diff__1,axiom,(
    ! [V_v] :
      ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_v),c_Int_OPls))
       => hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )
      & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_v),c_Int_OPls))
       => hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_Opred(V_v)) ) ) )).

fof(fact_setsum__bounded,axiom,(
    ! [V_K_2,V_f_2,V_A_2,T_b,T_c] :
      ( ( class_Groups_Oordered__ab__semigroup__add(T_c)
        & class_Rings_Osemiring__1(T_c) )
     => ( ! [B_i] :
            ( hBOOL(hAPP(hAPP(c_member(T_b),B_i),V_A_2))
           => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(V_f_2,B_i)),V_K_2)) )
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),V_A_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_c),hAPP(c_Finite__Set_Ocard(T_b),V_A_2))),V_K_2))) ) ) )).

fof(fact_Nats__number__of,axiom,(
    ! [V_w_2,T_b] :
      ( class_Int_Onumber__ring(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OPls),V_w_2))
       => hBOOL(hAPP(hAPP(c_member(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)),c_Nat_Osemiring__1__class_ONats(T_b))) ) ) )).

fof(fact_Suc__diff__number__of,axiom,(
    ! [V_m,V_v] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OPls),V_v))
     => hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_Opred(V_v))) ) )).

fof(fact_neg__number__of__pred__iff__0,axiom,(
    ! [V_v_2] :
      ( hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_Opred(V_v_2))))
    <=> hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )).

fof(fact_transfer__int__nat__relations_I1_J,axiom,(
    ! [V_y_2,V_x_2] :
      ( hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_x_2) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_y_2)
    <=> V_x_2 = V_y_2 ) )).

fof(fact_int__int__eq,axiom,(
    ! [V_n_2,V_m_2] :
      ( hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m_2) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n_2)
    <=> V_m_2 = V_n_2 ) )).

fof(fact_succ__pred,axiom,(
    ! [V_x] : c_Int_Osucc(c_Int_Opred(V_x)) = V_x )).

fof(fact_le__iff__pred__less,axiom,(
    ! [V_l_2,V_k_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k_2),V_l_2))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_Opred(V_k_2)),V_l_2)) ) )).

fof(fact_pred__def,axiom,(
    ! [V_k] : c_Int_Opred(V_k) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_k),c_Groups_Oone__class_Oone(tc_Int_Oint)) )).

fof(fact_Nats__0,axiom,(
    ! [T_b] :
      ( class_Rings_Osemiring__1(T_b)
     => hBOOL(hAPP(hAPP(c_member(T_b),c_Groups_Ozero__class_Ozero(T_b)),c_Nat_Osemiring__1__class_ONats(T_b))) ) )).

fof(fact_Nats__add,axiom,(
    ! [V_b_2,V_a_2,T_b] :
      ( class_Rings_Osemiring__1(T_b)
     => ( hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),c_Nat_Osemiring__1__class_ONats(T_b)))
       => ( hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),c_Nat_Osemiring__1__class_ONats(T_b)))
         => hBOOL(hAPP(hAPP(c_member(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_a_2),V_b_2)),c_Nat_Osemiring__1__class_ONats(T_b))) ) ) ) )).

fof(fact_Nats__mult,axiom,(
    ! [V_b_2,V_a_2,T_b] :
      ( class_Rings_Osemiring__1(T_b)
     => ( hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),c_Nat_Osemiring__1__class_ONats(T_b)))
       => ( hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),c_Nat_Osemiring__1__class_ONats(T_b)))
         => hBOOL(hAPP(hAPP(c_member(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_b_2)),c_Nat_Osemiring__1__class_ONats(T_b))) ) ) ) )).

fof(fact_Nats__1,axiom,(
    ! [T_b] :
      ( class_Rings_Osemiring__1(T_b)
     => hBOOL(hAPP(hAPP(c_member(T_b),c_Groups_Oone__class_Oone(T_b)),c_Nat_Osemiring__1__class_ONats(T_b))) ) )).

fof(fact_of__nat__in__Nats,axiom,(
    ! [V_n_2,T_b] :
      ( class_Rings_Osemiring__1(T_b)
     => hBOOL(hAPP(hAPP(c_member(T_b),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_b),V_n_2)),c_Nat_Osemiring__1__class_ONats(T_b))) ) )).

fof(fact_decr__mult__lemma,axiom,(
    ! [V_k_2,V_P_2,V_d_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_d_2))
     => ( ! [B_x] :
            ( hBOOL(hAPP(V_P_2,B_x))
           => hBOOL(hAPP(V_P_2,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B_x),V_d_2))) )
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_k_2))
         => ! [B_x] :
              ( hBOOL(hAPP(V_P_2,B_x))
             => hBOOL(hAPP(V_P_2,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_k_2),V_d_2)))) ) ) ) ) )).

fof(fact_incr__mult__lemma,axiom,(
    ! [V_k_2,V_P_2,V_d_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_d_2))
     => ( ! [B_x] :
            ( hBOOL(hAPP(V_P_2,B_x))
           => hBOOL(hAPP(V_P_2,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B_x),V_d_2))) )
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_k_2))
         => ! [B_x] :
              ( hBOOL(hAPP(V_P_2,B_x))
             => hBOOL(hAPP(V_P_2,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_k_2),V_d_2)))) ) ) ) ) )).

fof(fact_zero__less__imp__eq__int,axiom,(
    ! [V_k] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_k))
     => ? [B_n] :
          ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B_n))
          & V_k = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),B_n) ) ) )).

fof(fact_decr__lemma,axiom,(
    ! [V_z,V_x,V_d] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_d))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_x),V_z))),c_Groups_Oone__class_Oone(tc_Int_Oint))),V_d))),V_z)) ) )).

fof(fact_incr__lemma,axiom,(
    ! [V_x,V_z,V_d] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_d))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_z),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_x),V_z))),c_Groups_Oone__class_Oone(tc_Int_Oint))),V_d)))) ) )).

fof(fact_Suc__diff__eq__diff__pred,axiom,(
    ! [V_m,V_n] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OPls)),V_n))
     => hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),V_n) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(c_Int_OPls)))) ) )).

fof(fact_bool_Osize_I2_J,axiom,(
    c_HOL_Obool_Obool__size(c_fFalse) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )).

fof(fact_abs__idempotent,axiom,(
    ! [V_a,T_a] :
      ( class_Groups_Oordered__ab__group__add__abs(T_a)
     => hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)) = hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a) ) )).

fof(fact_rel__simps_I51_J,axiom,(
    ! [V_l_2,V_k_2] :
      ( c_Int_OBit1(V_k_2) = c_Int_OBit1(V_l_2)
    <=> V_k_2 = V_l_2 ) )).

fof(fact_abs__le__D1,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Groups_Oordered__ab__group__add__abs(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)),V_b))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b)) ) ) )).

fof(fact_abs__ge__self,axiom,(
    ! [V_a,T_a] :
      ( class_Groups_Oordered__ab__group__add__abs(T_a)
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a))) ) )).

fof(fact_abs__zero,axiom,(
    ! [T_a] :
      ( class_Groups_Oordered__ab__group__add__abs(T_a)
     => hAPP(c_Groups_Oabs__class_Oabs(T_a),c_Groups_Ozero__class_Ozero(T_a)) = c_Groups_Ozero__class_Ozero(T_a) ) )).

fof(fact_abs__eq__0,axiom,(
    ! [V_a_2,T_b] :
      ( class_Groups_Oordered__ab__group__add__abs(T_b)
     => ( hAPP(c_Groups_Oabs__class_Oabs(T_b),V_a_2) = c_Groups_Ozero__class_Ozero(T_b)
      <=> V_a_2 = c_Groups_Ozero__class_Ozero(T_b) ) ) )).

fof(fact_abs__add__abs,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Groups_Oordered__ab__group__add__abs(T_a)
     => hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_b))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_b)) ) )).

fof(fact_abs__mult__self,axiom,(
    ! [V_a,T_a] :
      ( class_Rings_Olinordered__idom(T_a)
     => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_a) ) )).

fof(fact_abs__mult,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Rings_Olinordered__idom(T_a)
     => hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_b)) ) )).

fof(fact_abs__minus__commute,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Groups_Oordered__ab__group__add__abs(T_a)
     => hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_b)) = hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_b),V_a)) ) )).

fof(fact_abs__one,axiom,(
    ! [T_a] :
      ( class_Rings_Olinordered__idom(T_a)
     => hAPP(c_Groups_Oabs__class_Oabs(T_a),c_Groups_Oone__class_Oone(T_a)) = c_Groups_Oone__class_Oone(T_a) ) )).

fof(fact_abs__of__nat,axiom,(
    ! [V_n,T_a] :
      ( class_Rings_Olinordered__idom(T_a)
     => hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),V_n)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),V_n) ) )).

fof(fact_abs__divide,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Fields_Olinordered__field__inverse__zero(T_a)
     => hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_b)) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_b)) ) )).

fof(fact_abs__int__eq,axiom,(
    ! [V_m] : hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m) )).

fof(fact_less__eq__int__code_I16_J,axiom,(
    ! [V_k2_2,V_k1_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit1(V_k1_2)),c_Int_OBit1(V_k2_2)))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k1_2),V_k2_2)) ) )).

fof(fact_rel__simps_I34_J,axiom,(
    ! [V_l_2,V_k_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit1(V_k_2)),c_Int_OBit1(V_l_2)))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k_2),V_l_2)) ) )).

fof(fact_less__int__code_I16_J,axiom,(
    ! [V_k2_2,V_k1_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit1(V_k1_2)),c_Int_OBit1(V_k2_2)))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k1_2),V_k2_2)) ) )).

fof(fact_rel__simps_I17_J,axiom,(
    ! [V_l_2,V_k_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit1(V_k_2)),c_Int_OBit1(V_l_2)))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),V_l_2)) ) )).

fof(fact_rel__simps_I39_J,axiom,(
    ! [V_l] : c_Int_OPls != c_Int_OBit1(V_l) )).

fof(fact_rel__simps_I46_J,axiom,(
    ! [V_k] : c_Int_OBit1(V_k) != c_Int_OPls )).

fof(fact_abs__of__nonneg,axiom,(
    ! [V_a,T_a] :
      ( class_Groups_Oordered__ab__group__add__abs(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a))
       => hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a) = V_a ) ) )).

fof(fact_abs__le__zero__iff,axiom,(
    ! [V_a_2,T_b] :
      ( class_Groups_Oordered__ab__group__add__abs(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Groups_Oabs__class_Oabs(T_b),V_a_2)),c_Groups_Ozero__class_Ozero(T_b)))
      <=> V_a_2 = c_Groups_Ozero__class_Ozero(T_b) ) ) )).

fof(fact_abs__ge__zero,axiom,(
    ! [V_a,T_a] :
      ( class_Groups_Oordered__ab__group__add__abs(T_a)
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a))) ) )).

fof(fact_abs__of__pos,axiom,(
    ! [V_a,T_a] :
      ( class_Groups_Oordered__ab__group__add__abs(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a))
       => hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a) = V_a ) ) )).

fof(fact_zero__less__abs__iff,axiom,(
    ! [V_a_2,T_b] :
      ( class_Groups_Oordered__ab__group__add__abs(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(c_Groups_Oabs__class_Oabs(T_b),V_a_2)))
      <=> V_a_2 != c_Groups_Ozero__class_Ozero(T_b) ) ) )).

fof(fact_abs__not__less__zero,axiom,(
    ! [V_a,T_a] :
      ( class_Groups_Oordered__ab__group__add__abs(T_a)
     => ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)),c_Groups_Ozero__class_Ozero(T_a))) ) )).

fof(fact_abs__triangle__ineq,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Groups_Oordered__ab__group__add__abs(T_a)
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b))),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_b)))) ) )).

fof(fact_abs__mult__less,axiom,(
    ! [V_d,V_b,V_c,V_a,T_a] :
      ( class_Rings_Olinordered__idom(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)),V_c))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_b)),V_d))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_b))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_d))) ) ) ) )).

fof(fact_abs__triangle__ineq3,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Groups_Oordered__ab__group__add__abs(T_a)
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_b)))),hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_b)))) ) )).

fof(fact_abs__triangle__ineq2,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Groups_Oordered__ab__group__add__abs(T_a)
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_b))),hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_b)))) ) )).

fof(fact_abs__triangle__ineq2__sym,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Groups_Oordered__ab__group__add__abs(T_a)
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_b))),hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_b),V_a)))) ) )).

fof(fact_nonzero__abs__divide,axiom,(
    ! [V_a,V_b,T_a] :
      ( class_Fields_Olinordered__field(T_a)
     => ( V_b != c_Groups_Ozero__class_Ozero(T_a)
       => hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_b)) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_b)) ) ) )).

fof(fact_bin__less__0__simps_I4_J,axiom,(
    ! [V_w_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit1(V_w_2)),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_w_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) ) )).

fof(fact_abs__zmult__eq__1,axiom,(
    ! [V_n,V_m] :
      ( hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_m),V_n)) = c_Groups_Oone__class_Oone(tc_Int_Oint)
     => hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_m) = c_Groups_Oone__class_Oone(tc_Int_Oint) ) )).

fof(fact_rel__simps_I22_J,axiom,(
    ! [V_k_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OPls),c_Int_OBit1(V_k_2)))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OPls),V_k_2)) ) )).

fof(fact_rel__simps_I12_J,axiom,(
    ! [V_k_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit1(V_k_2)),c_Int_OPls))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),c_Int_OPls)) ) )).

fof(fact_Bit1__def,axiom,(
    ! [V_k] : c_Int_OBit1(V_k) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),V_k)),V_k) )).

fof(fact_neg__number__of__Bit1,axiom,(
    ! [V_w_2] :
      ( hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit1(V_w_2))))
    <=> hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w_2))) ) )).

fof(fact_succ__Pls,axiom,(
    c_Int_Osucc(c_Int_OPls) = c_Int_OBit1(c_Int_OPls) )).

fof(fact_abs__eq__mult,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Rings_Oordered__ring__abs(T_a)
     => ( ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a))
            | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a))) )
          & ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_b))
            | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),c_Groups_Ozero__class_Ozero(T_a))) ) )
       => hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_b)) ) ) )).

fof(fact_abs__mult__pos,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Rings_Olinordered__idom(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_x))
       => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_y)),V_x) = hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_y),V_x)) ) ) )).

fof(fact_abs__triangle__ineq4,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Groups_Oordered__ab__group__add__abs(T_a)
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_b))),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_b)))) ) )).

fof(fact_abs__diff__triangle__ineq,axiom,(
    ! [V_d,V_c,V_b,V_a,T_a] :
      ( class_Groups_Oordered__ab__group__add__abs(T_a)
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),V_d)))),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_c))),hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_b),V_d))))) ) )).

fof(fact_abs__div__pos,axiom,(
    ! [V_x,V_y,T_a] :
      ( class_Fields_Olinordered__field__inverse__zero(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_y))
       => hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_x)),V_y) = hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_y)) ) ) )).

fof(fact_number__of__Bit1,axiom,(
    ! [V_w,T_a] :
      ( class_Int_Onumber__ring(T_a)
     => hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit1(V_w)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Groups_Oone__class_Oone(T_a)),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w))),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w)) ) )).

fof(fact_zabs__less__one__iff,axiom,(
    ! [V_z_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_z_2)),c_Groups_Oone__class_Oone(tc_Int_Oint)))
    <=> V_z_2 = c_Groups_Ozero__class_Ozero(tc_Int_Oint) ) )).

fof(fact_mult__numeral__1__right,axiom,(
    ! [V_a,T_a] :
      ( class_Int_Onumber__ring(T_a)
     => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit1(c_Int_OPls))) = V_a ) )).

fof(fact_mult__numeral__1,axiom,(
    ! [V_a,T_a] :
      ( class_Int_Onumber__ring(T_a)
     => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit1(c_Int_OPls))),V_a) = V_a ) )).

fof(fact_semiring__norm_I110_J,axiom,(
    ! [T_a] :
      ( class_Int_Onumber__ring(T_a)
     => c_Groups_Oone__class_Oone(T_a) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit1(c_Int_OPls)) ) )).

fof(fact_numeral__1__eq__1,axiom,(
    ! [T_a] :
      ( class_Int_Onumber__ring(T_a)
     => hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit1(c_Int_OPls)) = c_Groups_Oone__class_Oone(T_a) ) )).

fof(fact_rel__simps_I5_J,axiom,(
    ! [V_k_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OPls),c_Int_OBit1(V_k_2)))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OPls),V_k_2)) ) )).

fof(fact_rel__simps_I29_J,axiom,(
    ! [V_k_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit1(V_k_2)),c_Int_OPls))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),c_Int_OPls)) ) )).

fof(fact_divide__Numeral1,axiom,(
    ! [V_x,T_a] :
      ( ( class_Fields_Ofield(T_a)
        & class_Int_Onumber__ring(T_a) )
     => hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit1(c_Int_OPls))) = V_x ) )).

fof(fact_divide__numeral__1,axiom,(
    ! [V_a,T_a] :
      ( ( class_Fields_Ofield(T_a)
        & class_Int_Onumber__ring(T_a) )
     => hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit1(c_Int_OPls))) = V_a ) )).

fof(fact_one__is__num__one,axiom,(
    c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)) )).

fof(fact_Numeral1__eq1__nat,axiom,(
    c_Groups_Oone__class_Oone(tc_Nat_Onat) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(c_Int_OPls)) )).

fof(fact_nat__numeral__1__eq__1,axiom,(
    hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(c_Int_OPls)) = c_Groups_Oone__class_Oone(tc_Nat_Onat) )).

fof(fact_numeral__1__eq__Suc__0,axiom,(
    hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(c_Int_OPls)) = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) )).

fof(fact_numeral__3__eq__3,axiom,(
    hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(c_Int_OBit1(c_Int_OPls))) = hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) )).

fof(fact_Nat__Transfer_Otransfer__nat__int__function__closures_I8_J,axiom,(
    hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit1(c_Int_OBit1(c_Int_OPls))))) )).

fof(fact_Suc3__eq__add__3,axiom,(
    ! [V_n] : hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,V_n))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(c_Int_OBit1(c_Int_OPls)))),V_n) )).

fof(fact_transfer__int__nat__numerals_I4_J,axiom,(
    hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit1(c_Int_OBit1(c_Int_OPls))) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(c_Int_OBit1(c_Int_OPls)))) )).

fof(fact_le__special_I2_J,axiom,(
    ! [V_y_2,T_b] :
      ( ( class_Int_Onumber__ring(T_b)
        & class_Rings_Olinordered__idom(T_b) )
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Groups_Oone__class_Oone(T_b)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_y_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)),V_y_2)) ) ) )).

fof(fact_le__special_I4_J,axiom,(
    ! [V_x_2,T_b] :
      ( ( class_Int_Onumber__ring(T_b)
        & class_Rings_Olinordered__idom(T_b) )
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_x_2)),c_Groups_Oone__class_Oone(T_b)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_x_2),c_Int_OBit1(c_Int_OPls))) ) ) )).

fof(fact_less__special_I4_J,axiom,(
    ! [V_x_2,T_b] :
      ( ( class_Int_Onumber__ring(T_b)
        & class_Rings_Olinordered__idom(T_b) )
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_x_2)),c_Groups_Oone__class_Oone(T_b)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_x_2),c_Int_OBit1(c_Int_OPls))) ) ) )).

fof(fact_less__special_I2_J,axiom,(
    ! [V_y_2,T_b] :
      ( ( class_Int_Onumber__ring(T_b)
        & class_Rings_Olinordered__idom(T_b) )
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Oone__class_Oone(T_b)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_y_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)),V_y_2)) ) ) )).

fof(fact_add__special_I3_J,axiom,(
    ! [V_v,T_a] :
      ( class_Int_Onumber__ring(T_a)
     => hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_v)),c_Groups_Oone__class_Oone(T_a)) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_v),c_Int_OBit1(c_Int_OPls))) ) )).

fof(fact_add__special_I2_J,axiom,(
    ! [V_w,T_a] :
      ( class_Int_Onumber__ring(T_a)
     => hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Groups_Oone__class_Oone(T_a)),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w)) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)),V_w)) ) )).

fof(fact_bool_Osize_I1_J,axiom,(
    c_HOL_Obool_Obool__size(c_fTrue) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )).

fof(fact_one__code__numeral__code,axiom,(
    c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),c_Int_OBit1(c_Int_OPls)) )).

fof(fact_diff__special_I2_J,axiom,(
    ! [V_v,T_a] :
      ( class_Int_Onumber__ring(T_a)
     => hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_v)),c_Groups_Oone__class_Oone(T_a)) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_v),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)))) ) )).

fof(fact_diff__special_I1_J,axiom,(
    ! [V_w,T_a] :
      ( class_Int_Onumber__ring(T_a)
     => hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),c_Groups_Oone__class_Oone(T_a)),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w)) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_w))) ) )).

fof(fact_less__2__cases,axiom,(
    ! [V_n] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))
     => ( V_n = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
        | V_n = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) ) ) )).

fof(fact_half__gt__zero,axiom,(
    ! [V_r,T_a] :
      ( ( class_Fields_Olinordered__field__inverse__zero(T_a)
        & class_Int_Onumber__ring(T_a) )
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_r))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_r),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) ) ) )).

fof(fact_half__gt__zero__iff,axiom,(
    ! [V_r_2,T_b] :
      ( ( class_Fields_Olinordered__field__inverse__zero(T_b)
        & class_Int_Onumber__ring(T_b) )
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_r_2),hAPP(c_Int_Onumber__class_Onumber__of(T_b),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_r_2)) ) ) )).

fof(fact_int__val__lemma,axiom,(
    ! [V_k_2,V_f_2,V_n_2] :
      ( ! [B_i] :
          ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_i),V_n_2))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(V_f_2,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B_i),c_Groups_Oone__class_Oone(tc_Nat_Onat)))),hAPP(V_f_2,B_i)))),c_Groups_Oone__class_Oone(tc_Int_Oint))) )
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(V_f_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),V_k_2))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k_2),hAPP(V_f_2,V_n_2)))
         => ? [B_i] :
              ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B_i),V_n_2))
              & hAPP(V_f_2,B_i) = V_k_2 ) ) ) ) )).

fof(fact_negative__zle,axiom,(
    ! [V_m,V_n] : hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n))),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m))) )).

fof(fact_negative__zless,axiom,(
    ! [V_m,V_n] : hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Nat_OSuc,V_n)))),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m))) )).

fof(fact_arith__simps_I30_J,axiom,(
    ! [V_w,T_a] :
      ( class_Int_Onumber__ring(T_a)
     => hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w)) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_w)) ) )).

fof(fact_number__of__minus,axiom,(
    ! [V_w,T_a] :
      ( class_Int_Onumber__ring(T_a)
     => hAPP(c_Int_Onumber__class_Onumber__of(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_w)) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w)) ) )).

fof(fact_code__numeral__zero__minus__one,axiom,(
    hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)) = c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) )).

fof(fact_rel__simps_I48_J,axiom,(
    ! [V_l_2,V_k_2] :
      ( c_Int_OBit0(V_k_2) = c_Int_OBit0(V_l_2)
    <=> V_k_2 = V_l_2 ) )).

fof(fact_minus__Bit0,axiom,(
    ! [V_k] : hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Int_OBit0(V_k)) = c_Int_OBit0(hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_k)) )).

fof(fact_zminus__zminus,axiom,(
    ! [V_z] : hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_z)) = V_z )).

fof(fact_compl__eq__compl__iff,axiom,(
    ! [V_y_2,V_x_2,T_b] :
      ( class_Lattices_Oboolean__algebra(T_b)
     => ( hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_x_2) = hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_y_2)
      <=> V_x_2 = V_y_2 ) ) )).

fof(fact_neg__equal__iff__equal,axiom,(
    ! [V_b_2,V_a_2,T_b] :
      ( class_Groups_Ogroup__add(T_b)
     => ( hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_a_2) = hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_b_2)
      <=> V_a_2 = V_b_2 ) ) )).

fof(fact_minus__equation__iff,axiom,(
    ! [V_b_2,V_a_2,T_b] :
      ( class_Groups_Ogroup__add(T_b)
     => ( hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_a_2) = V_b_2
      <=> hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_b_2) = V_a_2 ) ) )).

fof(fact_equation__minus__iff,axiom,(
    ! [V_b_2,V_a_2,T_b] :
      ( class_Groups_Ogroup__add(T_b)
     => ( V_a_2 = hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_b_2)
      <=> V_b_2 = hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_a_2) ) ) )).

fof(fact_uminus__apply,axiom,(
    ! [V_x_2,V_A_2,T_c,T_b] :
      ( class_Groups_Ouminus(T_b)
     => hAPP(hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,T_b)),V_A_2),V_x_2) = hAPP(c_Groups_Ouminus__class_Ouminus(T_b),hAPP(V_A_2,V_x_2)) ) )).

fof(fact_double__compl,axiom,(
    ! [V_x,T_a] :
      ( class_Lattices_Oboolean__algebra(T_a)
     => hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_x)) = V_x ) )).

fof(fact_minus__minus,axiom,(
    ! [V_a,T_a] :
      ( class_Groups_Ogroup__add(T_a)
     => hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)) = V_a ) )).

fof(fact_minus__number__of__mult,axiom,(
    ! [V_z,V_w,T_a] :
      ( class_Int_Onumber__ring(T_a)
     => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w))),V_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_w))),V_z) ) )).

fof(fact_le__minus__iff,axiom,(
    ! [V_b_2,V_a_2,T_b] :
      ( class_Groups_Oordered__ab__group__add(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_b_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_b_2),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_a_2))) ) ) )).

fof(fact_minus__le__iff,axiom,(
    ! [V_b_2,V_a_2,T_b] :
      ( class_Groups_Oordered__ab__group__add(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_a_2)),V_b_2))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_b_2)),V_a_2)) ) ) )).

fof(fact_neg__le__iff__le,axiom,(
    ! [V_a_2,V_b_2,T_b] :
      ( class_Groups_Oordered__ab__group__add(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_b_2)),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_a_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),V_b_2)) ) ) )).

fof(fact_compl__le__compl__iff,axiom,(
    ! [V_y_2,V_x_2,T_b] :
      ( class_Lattices_Oboolean__algebra(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_x_2)),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_y_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_y_2),V_x_2)) ) ) )).

fof(fact_le__imp__neg__le,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Groups_Oordered__ab__group__add(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_b)),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a))) ) ) )).

fof(fact_compl__mono,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Lattices_Oboolean__algebra(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_y)),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_x))) ) ) )).

fof(fact_neg__equal__zero,axiom,(
    ! [V_a_2,T_b] :
      ( class_Groups_Olinordered__ab__group__add(T_b)
     => ( hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_a_2) = V_a_2
      <=> V_a_2 = c_Groups_Ozero__class_Ozero(T_b) ) ) )).

fof(fact_neg__equal__0__iff__equal,axiom,(
    ! [V_a_2,T_b] :
      ( class_Groups_Ogroup__add(T_b)
     => ( hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_a_2) = c_Groups_Ozero__class_Ozero(T_b)
      <=> V_a_2 = c_Groups_Ozero__class_Ozero(T_b) ) ) )).

fof(fact_equal__neg__zero,axiom,(
    ! [V_a_2,T_b] :
      ( class_Groups_Olinordered__ab__group__add(T_b)
     => ( V_a_2 = hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_a_2)
      <=> V_a_2 = c_Groups_Ozero__class_Ozero(T_b) ) ) )).

fof(fact_neg__0__equal__iff__equal,axiom,(
    ! [V_a_2,T_b] :
      ( class_Groups_Ogroup__add(T_b)
     => ( c_Groups_Ozero__class_Ozero(T_b) = hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_a_2)
      <=> c_Groups_Ozero__class_Ozero(T_b) = V_a_2 ) ) )).

fof(fact_minus__zero,axiom,(
    ! [T_a] :
      ( class_Groups_Ogroup__add(T_a)
     => hAPP(c_Groups_Ouminus__class_Ouminus(T_a),c_Groups_Ozero__class_Ozero(T_a)) = c_Groups_Ozero__class_Ozero(T_a) ) )).

fof(fact_neg__less__iff__less,axiom,(
    ! [V_a_2,V_b_2,T_b] :
      ( class_Groups_Oordered__ab__group__add(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_b_2)),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_a_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),V_b_2)) ) ) )).

fof(fact_minus__less__iff,axiom,(
    ! [V_b_2,V_a_2,T_b] :
      ( class_Groups_Oordered__ab__group__add(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_a_2)),V_b_2))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_b_2)),V_a_2)) ) ) )).

fof(fact_less__minus__iff,axiom,(
    ! [V_b_2,V_a_2,T_b] :
      ( class_Groups_Oordered__ab__group__add(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_b_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_b_2),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_a_2))) ) ) )).

fof(fact_minus__add__cancel,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Groups_Ogroup__add(T_a)
     => hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)) = V_b ) )).

fof(fact_add__minus__cancel,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Groups_Ogroup__add(T_a)
     => hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),V_b)) = V_b ) )).

fof(fact_minus__add,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Groups_Ogroup__add(T_a)
     => hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_b)),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)) ) )).

fof(fact_minus__add__distrib,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Groups_Oab__group__add(T_a)
     => hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_b)) ) )).

fof(fact_minus__mult__right,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Rings_Oring(T_a)
     => hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_b)) ) )).

fof(fact_minus__mult__left,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Rings_Oring(T_a)
     => hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),V_b) ) )).

fof(fact_minus__mult__commute,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Rings_Oring(T_a)
     => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),V_b) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_b)) ) )).

fof(fact_minus__mult__minus,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Rings_Oring(T_a)
     => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_b)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b) ) )).

fof(fact_square__eq__iff,axiom,(
    ! [V_b_2,V_a_2,T_b] :
      ( class_Rings_Oidom(T_b)
     => ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_a_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_b_2),V_b_2)
      <=> ( V_a_2 = V_b_2
          | V_a_2 = hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_b_2) ) ) ) )).

fof(fact_minus__diff__eq,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Groups_Oab__group__add(T_a)
     => hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_b)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_b),V_a) ) )).

fof(fact_rel__simps_I31_J,axiom,(
    ! [V_l_2,V_k_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit0(V_k_2)),c_Int_OBit0(V_l_2)))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k_2),V_l_2)) ) )).

fof(fact_less__eq__int__code_I13_J,axiom,(
    ! [V_k2_2,V_k1_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit0(V_k1_2)),c_Int_OBit0(V_k2_2)))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k1_2),V_k2_2)) ) )).

fof(fact_rel__simps_I14_J,axiom,(
    ! [V_l_2,V_k_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit0(V_k_2)),c_Int_OBit0(V_l_2)))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),V_l_2)) ) )).

fof(fact_less__int__code_I13_J,axiom,(
    ! [V_k2_2,V_k1_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit0(V_k1_2)),c_Int_OBit0(V_k2_2)))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k1_2),V_k2_2)) ) )).

fof(fact_zminus__0,axiom,(
    hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)) = c_Groups_Ozero__class_Ozero(tc_Int_Oint) )).

fof(fact_rel__simps_I44_J,axiom,(
    ! [V_k_2] :
      ( c_Int_OBit0(V_k_2) = c_Int_OPls
    <=> V_k_2 = c_Int_OPls ) )).

fof(fact_rel__simps_I38_J,axiom,(
    ! [V_l_2] :
      ( c_Int_OPls = c_Int_OBit0(V_l_2)
    <=> c_Int_OPls = V_l_2 ) )).

fof(fact_Bit0__Pls,axiom,(
    c_Int_OBit0(c_Int_OPls) = c_Int_OPls )).

fof(fact_rel__simps_I50_J,axiom,(
    ! [V_l,V_k] : c_Int_OBit1(V_k) != c_Int_OBit0(V_l) )).

fof(fact_rel__simps_I49_J,axiom,(
    ! [V_l,V_k] : c_Int_OBit0(V_k) != c_Int_OBit1(V_l) )).

fof(fact_add__Bit0__Bit0,axiom,(
    ! [V_l,V_k] : hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Int_OBit0(V_k)),c_Int_OBit0(V_l)) = c_Int_OBit0(hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_k),V_l)) )).

fof(fact_Bit0__def,axiom,(
    ! [V_k] : c_Int_OBit0(V_k) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_k),V_k) )).

fof(fact_minus__divide__left,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Rings_Odivision__ring(T_a)
     => hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_b)) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),V_b) ) )).

fof(fact_minus__divide__divide,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Fields_Ofield__inverse__zero(T_a)
     => hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_b)) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_b) ) )).

fof(fact_minus__divide__right,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Fields_Ofield__inverse__zero(T_a)
     => hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_b)) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_b)) ) )).

fof(fact_mult__Bit0,axiom,(
    ! [V_l,V_k] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Int_OBit0(V_k)),V_l) = c_Int_OBit0(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_k),V_l)) )).

fof(fact_abs__minus__cancel,axiom,(
    ! [V_a,T_a] :
      ( class_Groups_Oordered__ab__group__add__abs(T_a)
     => hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)) = hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a) ) )).

fof(fact_minus__Pls,axiom,(
    hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Int_OPls) = c_Int_OPls )).

fof(fact_zminus__zadd__distrib,axiom,(
    ! [V_w,V_z] : hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_z),V_w)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_z)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_w)) )).

fof(fact_minus__numeral__code_I5_J,axiom,(
    ! [V_w] : hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_w)) )).

fof(fact_diff__bin__simps_I7_J,axiom,(
    ! [V_l,V_k] : hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Int_OBit0(V_k)),c_Int_OBit0(V_l)) = c_Int_OBit0(hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_k),V_l)) )).

fof(fact_zmult__zminus,axiom,(
    ! [V_w,V_z] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_z)),V_w) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_z),V_w)) )).

fof(fact_minus__le__self__iff,axiom,(
    ! [V_a_2,T_b] :
      ( class_Groups_Olinordered__ab__group__add(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_a_2)),V_a_2))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_a_2)) ) ) )).

fof(fact_neg__le__0__iff__le,axiom,(
    ! [V_a_2,T_b] :
      ( class_Groups_Oordered__ab__group__add(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_a_2)),c_Groups_Ozero__class_Ozero(T_b)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_a_2)) ) ) )).

fof(fact_le__minus__self__iff,axiom,(
    ! [V_a_2,T_b] :
      ( class_Groups_Olinordered__ab__group__add(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_a_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),c_Groups_Ozero__class_Ozero(T_b))) ) ) )).

fof(fact_neg__0__le__iff__le,axiom,(
    ! [V_a_2,T_b] :
      ( class_Groups_Oordered__ab__group__add(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_a_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),c_Groups_Ozero__class_Ozero(T_b))) ) ) )).

fof(fact_less__minus__self__iff,axiom,(
    ! [V_a_2,T_b] :
      ( class_Rings_Olinordered__idom(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_a_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),c_Groups_Ozero__class_Ozero(T_b))) ) ) )).

fof(fact_neg__0__less__iff__less,axiom,(
    ! [V_a_2,T_b] :
      ( class_Groups_Oordered__ab__group__add(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_a_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),c_Groups_Ozero__class_Ozero(T_b))) ) ) )).

fof(fact_neg__less__0__iff__less,axiom,(
    ! [V_a_2,T_b] :
      ( class_Groups_Oordered__ab__group__add(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_a_2)),c_Groups_Ozero__class_Ozero(T_b)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_a_2)) ) ) )).

fof(fact_neg__less__nonneg,axiom,(
    ! [V_a_2,T_b] :
      ( class_Groups_Olinordered__ab__group__add(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_a_2)),V_a_2))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_a_2)) ) ) )).

fof(fact_minus__unique,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Groups_Ogroup__add(T_a)
     => ( hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b) = c_Groups_Ozero__class_Ozero(T_a)
       => hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a) = V_b ) ) )).

fof(fact_ab__left__minus,axiom,(
    ! [V_a,T_a] :
      ( class_Groups_Oab__group__add(T_a)
     => hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),V_a) = c_Groups_Ozero__class_Ozero(T_a) ) )).

fof(fact_left__minus,axiom,(
    ! [V_a,T_a] :
      ( class_Groups_Ogroup__add(T_a)
     => hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),V_a) = c_Groups_Ozero__class_Ozero(T_a) ) )).

fof(fact_eq__neg__iff__add__eq__0,axiom,(
    ! [V_b_2,V_a_2,T_b] :
      ( class_Groups_Ogroup__add(T_b)
     => ( V_a_2 = hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_b_2)
      <=> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_a_2),V_b_2) = c_Groups_Ozero__class_Ozero(T_b) ) ) )).

fof(fact_right__minus,axiom,(
    ! [V_a,T_a] :
      ( class_Groups_Ogroup__add(T_a)
     => hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)) = c_Groups_Ozero__class_Ozero(T_a) ) )).

fof(fact_diff__0,axiom,(
    ! [V_a,T_a] :
      ( class_Groups_Ogroup__add(T_a)
     => hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a) ) )).

fof(fact_comm__ring__1__class_Onormalizing__ring__rules_I2_J,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Rings_Ocomm__ring__1(T_a)
     => hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_x),V_y) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_x),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_y)) ) )).

fof(fact_diff__minus__eq__add,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Groups_Ogroup__add(T_a)
     => hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_b)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b) ) )).

fof(fact_ab__diff__minus,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Groups_Oab__group__add(T_a)
     => hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_b) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_b)) ) )).

fof(fact_diff__def,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Groups_Ogroup__add(T_a)
     => hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_b) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_b)) ) )).

fof(fact_comm__ring__1__class_Onormalizing__ring__rules_I1_J,axiom,(
    ! [V_x,T_a] :
      ( class_Rings_Ocomm__ring__1(T_a)
     => hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),c_Groups_Oone__class_Oone(T_a))),V_x) ) )).

fof(fact_square__eq__1__iff,axiom,(
    ! [V_x_2,T_b] :
      ( class_Rings_Oring__1__no__zero__divisors(T_b)
     => ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_x_2),V_x_2) = c_Groups_Oone__class_Oone(T_b)
      <=> ( V_x_2 = c_Groups_Oone__class_Oone(T_b)
          | V_x_2 = hAPP(c_Groups_Ouminus__class_Ouminus(T_b),c_Groups_Oone__class_Oone(T_b)) ) ) ) )).

fof(fact_nonzero__minus__divide__divide,axiom,(
    ! [V_a,V_b,T_a] :
      ( class_Rings_Odivision__ring(T_a)
     => ( V_b != c_Groups_Ozero__class_Ozero(T_a)
       => hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_b)) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_b) ) ) )).

fof(fact_nonzero__minus__divide__right,axiom,(
    ! [V_a,V_b,T_a] :
      ( class_Rings_Odivision__ring(T_a)
     => ( V_b != c_Groups_Ozero__class_Ozero(T_a)
       => hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_b)) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_b)) ) ) )).

fof(fact_abs__le__D2,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Groups_Oordered__ab__group__add__abs(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)),V_b))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),V_b)) ) ) )).

fof(fact_abs__leI,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Groups_Oordered__ab__group__add__abs(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),V_b))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)),V_b)) ) ) ) )).

fof(fact_abs__le__iff,axiom,(
    ! [V_b_2,V_a_2,T_b] :
      ( class_Groups_Oordered__ab__group__add__abs(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Groups_Oabs__class_Oabs(T_b),V_a_2)),V_b_2))
      <=> ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),V_b_2))
          & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_a_2)),V_b_2)) ) ) ) )).

fof(fact_abs__ge__minus__self,axiom,(
    ! [V_a,T_a] :
      ( class_Groups_Oordered__ab__group__add__abs(T_a)
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a))) ) )).

fof(fact_abs__less__iff,axiom,(
    ! [V_b_2,V_a_2,T_b] :
      ( class_Rings_Olinordered__idom(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(c_Groups_Oabs__class_Oabs(T_b),V_a_2)),V_b_2))
      <=> ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),V_b_2))
          & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_a_2)),V_b_2)) ) ) ) )).

fof(fact_bin__less__0__simps_I3_J,axiom,(
    ! [V_w_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit0(V_w_2)),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_w_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) ) )).

fof(fact_rel__simps_I27_J,axiom,(
    ! [V_k_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit0(V_k_2)),c_Int_OPls))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k_2),c_Int_OPls)) ) )).

fof(fact_rel__simps_I21_J,axiom,(
    ! [V_k_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OPls),c_Int_OBit0(V_k_2)))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OPls),V_k_2)) ) )).

fof(fact_less__eq__int__code_I14_J,axiom,(
    ! [V_k2_2,V_k1_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit0(V_k1_2)),c_Int_OBit1(V_k2_2)))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k1_2),V_k2_2)) ) )).

fof(fact_rel__simps_I32_J,axiom,(
    ! [V_l_2,V_k_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit0(V_k_2)),c_Int_OBit1(V_l_2)))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k_2),V_l_2)) ) )).

fof(fact_rel__simps_I10_J,axiom,(
    ! [V_k_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit0(V_k_2)),c_Int_OPls))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),c_Int_OPls)) ) )).

fof(fact_rel__simps_I4_J,axiom,(
    ! [V_k_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OPls),c_Int_OBit0(V_k_2)))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OPls),V_k_2)) ) )).

fof(fact_less__int__code_I15_J,axiom,(
    ! [V_k2_2,V_k1_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit1(V_k1_2)),c_Int_OBit0(V_k2_2)))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k1_2),V_k2_2)) ) )).

fof(fact_rel__simps_I16_J,axiom,(
    ! [V_l_2,V_k_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit1(V_k_2)),c_Int_OBit0(V_l_2)))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),V_l_2)) ) )).

fof(fact_add__Bit0__Bit1,axiom,(
    ! [V_l,V_k] : hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Int_OBit0(V_k)),c_Int_OBit1(V_l)) = c_Int_OBit1(hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_k),V_l)) )).

fof(fact_add__Bit1__Bit0,axiom,(
    ! [V_l,V_k] : hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Int_OBit1(V_k)),c_Int_OBit0(V_l)) = c_Int_OBit1(hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_k),V_l)) )).

fof(fact_minus__min__eq__max,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Groups_Olinordered__ab__group__add(T_a)
     => hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),V_y)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_x)),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_y)) ) )).

fof(fact_minus__max__eq__min,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Groups_Olinordered__ab__group__add(T_a)
     => hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),V_y)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_x)),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_y)) ) )).

fof(fact_zadd__zminus__inverse2,axiom,(
    ! [V_z] : hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_z)),V_z) = c_Groups_Ozero__class_Ozero(tc_Int_Oint) )).

fof(fact_negative__eq__positive,axiom,(
    ! [V_m_2,V_n_2] :
      ( hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n_2)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m_2)
    <=> ( V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
        & V_m_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) ) )).

fof(fact_diff__bin__simps_I3_J,axiom,(
    ! [V_l] : hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Int_OPls),c_Int_OBit0(V_l)) = c_Int_OBit0(hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Int_OPls),V_l)) )).

fof(fact_diff__bin__simps_I10_J,axiom,(
    ! [V_l,V_k] : hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Int_OBit1(V_k)),c_Int_OBit1(V_l)) = c_Int_OBit0(hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_k),V_l)) )).

fof(fact_diff__bin__simps_I9_J,axiom,(
    ! [V_l,V_k] : hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Int_OBit1(V_k)),c_Int_OBit0(V_l)) = c_Int_OBit1(hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_k),V_l)) )).

fof(fact_not__int__zless__negative,axiom,(
    ! [V_m,V_n] : ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m)))) )).

fof(fact_diff__int__def__symmetric,axiom,(
    ! [V_w,V_z] : hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_z),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_w)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_z),V_w) )).

fof(fact_diff__int__def,axiom,(
    ! [V_w,V_z] : hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_z),V_w) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_z),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_w)) )).

fof(fact_Ints__minus,axiom,(
    ! [V_a_2,T_b] :
      ( class_Rings_Oring__1(T_b)
     => ( hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),c_Int_Oring__1__class_OInts(T_b)))
       => hBOOL(hAPP(hAPP(c_member(T_b),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_a_2)),c_Int_Oring__1__class_OInts(T_b))) ) ) )).

fof(fact_neg__number__of__Bit0,axiom,(
    ! [V_w_2] :
      ( hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(V_w_2))))
    <=> hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w_2))) ) )).

fof(fact_pred__Bit1,axiom,(
    ! [V_k] : c_Int_Opred(c_Int_OBit1(V_k)) = c_Int_OBit0(V_k) )).

fof(fact_pred__Bit0,axiom,(
    ! [V_k] : c_Int_Opred(c_Int_OBit0(V_k)) = c_Int_OBit1(c_Int_Opred(V_k)) )).

fof(fact_minus__Bit1,axiom,(
    ! [V_k] : hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Int_OBit1(V_k)) = c_Int_OBit1(c_Int_Opred(hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_k))) )).

fof(fact_succ__Bit0,axiom,(
    ! [V_k] : c_Int_Osucc(c_Int_OBit0(V_k)) = c_Int_OBit1(V_k) )).

fof(fact_succ__Bit1,axiom,(
    ! [V_k] : c_Int_Osucc(c_Int_OBit1(V_k)) = c_Int_OBit0(c_Int_Osucc(V_k)) )).

fof(fact_number__of__Bit0,axiom,(
    ! [V_w,T_a] :
      ( class_Int_Onumber__ring(T_a)
     => hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit0(V_w)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w))),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w)) ) )).

fof(fact_abs__minus__le__zero,axiom,(
    ! [V_a,T_a] :
      ( class_Groups_Oordered__ab__group__add__abs(T_a)
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a))),c_Groups_Ozero__class_Ozero(T_a))) ) )).

fof(fact_abs__of__nonpos,axiom,(
    ! [V_a,T_a] :
      ( class_Groups_Oordered__ab__group__add__abs(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a)))
       => hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a) ) ) )).

fof(fact_abs__if,axiom,(
    ! [V_a,T_a] :
      ( class_Groups_Oabs__if(T_a)
     => ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a)))
         => hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a) )
        & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a)))
         => hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a) = V_a ) ) ) )).

fof(fact_abs__of__neg,axiom,(
    ! [V_a,T_a] :
      ( class_Groups_Oordered__ab__group__add__abs(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a)))
       => hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a) ) ) )).

fof(fact_less__eq__int__code_I15_J,axiom,(
    ! [V_k2_2,V_k1_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit1(V_k1_2)),c_Int_OBit0(V_k2_2)))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k1_2),V_k2_2)) ) )).

fof(fact_rel__simps_I33_J,axiom,(
    ! [V_l_2,V_k_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit1(V_k_2)),c_Int_OBit0(V_l_2)))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),V_l_2)) ) )).

fof(fact_less__int__code_I14_J,axiom,(
    ! [V_k2_2,V_k1_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit0(V_k1_2)),c_Int_OBit1(V_k2_2)))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k1_2),V_k2_2)) ) )).

fof(fact_rel__simps_I15_J,axiom,(
    ! [V_l_2,V_k_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit0(V_k_2)),c_Int_OBit1(V_l_2)))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k_2),V_l_2)) ) )).

fof(fact_diff__number__of__eq,axiom,(
    ! [V_w,V_v,T_a] :
      ( class_Int_Onumber__ring(T_a)
     => hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w)) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_v),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_w))) ) )).

fof(fact_mult__Bit1,axiom,(
    ! [V_l,V_k] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Int_OBit1(V_k)),V_l) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Int_OBit0(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_k),V_l))),V_l) )).

fof(fact_int__zle__neg,axiom,(
    ! [V_m_2,V_n_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n_2)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m_2))))
    <=> ( V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
        & V_m_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) ) )).

fof(fact_negative__zle__0,axiom,(
    ! [V_n] : hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n))),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) )).

fof(fact_minus__numeral__code_I6_J,axiom,(
    ! [V_w,V_v] : hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_v),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_w))) )).

fof(fact_zabs__def,axiom,(
    ! [V_i] :
      ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_i),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
       => hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_i) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_i) )
      & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_i),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
       => hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_i) = V_i ) ) )).

fof(fact_neg__zminus__int,axiom,(
    ! [V_n] : hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Nat_OSuc,V_n))))) )).

fof(fact_diff__bin__simps_I8_J,axiom,(
    ! [V_l,V_k] : hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Int_OBit0(V_k)),c_Int_OBit1(V_l)) = c_Int_OBit1(hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Int_Opred(V_k)),V_l)) )).

fof(fact_add__Bit1__Bit1,axiom,(
    ! [V_l,V_k] : hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Int_OBit1(V_k)),c_Int_OBit1(V_l)) = c_Int_OBit0(hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_k),c_Int_Osucc(V_l))) )).

fof(fact_zero__code__numeral__code,axiom,(
    c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),c_Int_OPls) )).

fof(fact_double__number__of__Bit0,axiom,(
    ! [V_w,T_a] :
      ( class_Int_Onumber__ring(T_a)
     => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Groups_Oone__class_Oone(T_a)),c_Groups_Oone__class_Oone(T_a))),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w)) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit0(V_w)) ) )).

fof(fact_abs__number__of,axiom,(
    ! [V_x,T_a] :
      ( ( class_Int_Onumber__ring(T_a)
        & class_Rings_Olinordered__idom(T_a) )
     => ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_x)),c_Groups_Ozero__class_Ozero(T_a)))
         => hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_x)) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_x)) )
        & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_x)),c_Groups_Ozero__class_Ozero(T_a)))
         => hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_x)) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_x) ) ) ) )).

fof(fact_add__number__of__diff2,axiom,(
    ! [V_w,V_c,V_v,T_a] :
      ( class_Int_Onumber__ring(T_a)
     => hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_v)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_c),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_v),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_w)))),V_c) ) )).

fof(fact_number__of1,axiom,(
    ! [V_n] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_n)))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(V_n))))
        & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit1(V_n)))) ) ) )).

fof(fact_number__of__pred,axiom,(
    ! [V_w,T_a] :
      ( class_Int_Onumber__ring(T_a)
     => hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_Opred(V_w)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),c_Groups_Oone__class_Oone(T_a))),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w)) ) )).

fof(fact_not__zle__0__negative,axiom,(
    ! [V_n] : ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Nat_OSuc,V_n))))) )).

fof(fact_negative__zless__0,axiom,(
    ! [V_n] : hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Nat_OSuc,V_n)))),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) )).

fof(fact_mult__2__right,axiom,(
    ! [V_z,T_a] :
      ( class_Int_Onumber__ring(T_a)
     => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_z),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_z),V_z) ) )).

fof(fact_mult__2,axiom,(
    ! [V_z,T_a] :
      ( class_Int_Onumber__ring(T_a)
     => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_z) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_z),V_z) ) )).

fof(fact_one__add__one__is__two,axiom,(
    ! [T_a] :
      ( class_Int_Onumber__ring(T_a)
     => hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Groups_Oone__class_Oone(T_a)),c_Groups_Oone__class_Oone(T_a)) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))) ) )).

fof(fact_numeral__2__eq__2,axiom,(
    hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))) = hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) )).

fof(fact_semiring__norm_I115_J,axiom,(
    hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))) )).

fof(fact_Nat__Transfer_Otransfer__nat__int__function__closures_I7_J,axiom,(
    hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) )).

fof(fact_add__2__eq__Suc_H,axiom,(
    ! [V_n] : hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,V_n)) )).

fof(fact_add__2__eq__Suc,axiom,(
    ! [V_n] : hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_n) = hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,V_n)) )).

fof(fact_nat__mult__2__right,axiom,(
    ! [V_z] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_z),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_z),V_z) )).

fof(fact_nat__mult__2,axiom,(
    ! [V_z] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_z) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_z),V_z) )).

fof(fact_transfer__int__nat__numerals_I3_J,axiom,(
    hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) )).

fof(fact_nat__1__add__1,axiom,(
    hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))) )).

fof(fact_eq__special_I4_J,axiom,(
    ! [V_x_2,T_b] :
      ( class_Int_Onumber__ring(T_b)
     => ( hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_x_2) = c_Groups_Oone__class_Oone(T_b)
      <=> c_Int_Oiszero(T_b,hAPP(c_Int_Onumber__class_Onumber__of(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_x_2),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Int_OBit1(c_Int_OPls))))) ) ) )).

fof(fact_eq__special_I2_J,axiom,(
    ! [V_y_2,T_b] :
      ( class_Int_Onumber__ring(T_b)
     => ( c_Groups_Oone__class_Oone(T_b) = hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_y_2)
      <=> c_Int_Oiszero(T_b,hAPP(c_Int_Onumber__class_Onumber__of(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_y_2)))) ) ) )).

fof(fact_neg__zmod__mult__2,axiom,(
    ! [V_b,V_a] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_a),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
     => c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_b)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_a)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_b),c_Groups_Oone__class_Oone(tc_Int_Oint)),V_a))),c_Groups_Oone__class_Oone(tc_Int_Oint)) ) )).

fof(fact_zmod__number__of__Bit1,axiom,(
    ! [V_v,V_w] :
      ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)))
       => c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit1(V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(V_w))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)))),c_Groups_Oone__class_Oone(tc_Int_Oint)) )
      & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)))
       => c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit1(V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(V_w))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)))),c_Groups_Oone__class_Oone(tc_Int_Oint)) ) ) )).

fof(fact_pos__zmod__mult__2,axiom,(
    ! [V_b,V_a] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a))
     => c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_b)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_a)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_b,V_a))) ) )).

fof(fact_pos__zdiv__mult__2,axiom,(
    ! [V_b,V_a] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a))
     => hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_b))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_a)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_b),V_a) ) )).

fof(fact_neg__zdiv__mult__2,axiom,(
    ! [V_b,V_a] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_a),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
     => hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_b))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_a)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_b),c_Groups_Oone__class_Oone(tc_Int_Oint))),V_a) ) )).

fof(fact_ComplI,axiom,(
    ! [V_A_2,V_c_2,T_b] :
      ( ~ hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_A_2))
     => hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),V_A_2))) ) )).

fof(fact_DIVISION__BY__ZERO,axiom,(
    ! [V_a] :
      ( hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),c_Groups_Ozero__class_Ozero(tc_Int_Oint)) = c_Groups_Ozero__class_Ozero(tc_Int_Oint)
      & c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) = V_a ) )).

fof(fact_mod__div__trivial,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Divides_Osemiring__div(T_a)
     => hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),c_Divides_Odiv__class_Omod(T_a,V_a,V_b)),V_b) = c_Groups_Ozero__class_Ozero(T_a) ) )).

fof(fact_zmod__zdiv__trivial,axiom,(
    ! [V_b,V_a] : hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b)),V_b) = c_Groups_Ozero__class_Ozero(tc_Int_Oint) )).

fof(fact_mod__mod__trivial,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Divides_Osemiring__div(T_a)
     => c_Divides_Odiv__class_Omod(T_a,c_Divides_Odiv__class_Omod(T_a,V_a,V_b),V_b) = c_Divides_Odiv__class_Omod(T_a,V_a,V_b) ) )).

fof(fact_zdiv__zadd1__eq,axiom,(
    ! [V_c,V_b,V_a] : hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a),V_b)),V_c) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_c)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_b),V_c))),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_c)),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_b,V_c))),V_c)) )).

fof(fact_semiring__div__class_Omod__div__equality_H,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Divides_Osemiring__div(T_a)
     => hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Divides_Odiv__class_Omod(T_a,V_a,V_b)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_a),V_b)),V_b)) = V_a ) )).

fof(fact_mod__div__equality2,axiom,(
    ! [V_a,V_b,T_a] :
      ( class_Divides_Osemiring__div(T_a)
     => hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_a),V_b))),c_Divides_Odiv__class_Omod(T_a,V_a,V_b)) = V_a ) )).

fof(fact_mod__div__equality,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Divides_Osemiring__div(T_a)
     => hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_a),V_b)),V_b)),c_Divides_Odiv__class_Omod(T_a,V_a,V_b)) = V_a ) )).

fof(fact_div__mod__equality2,axiom,(
    ! [V_c,V_a,V_b,T_a] :
      ( class_Divides_Osemiring__div(T_a)
     => hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_a),V_b))),c_Divides_Odiv__class_Omod(T_a,V_a,V_b))),V_c) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c) ) )).

fof(fact_div__mod__equality,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Divides_Osemiring__div(T_a)
     => hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_a),V_b)),V_b)),c_Divides_Odiv__class_Omod(T_a,V_a,V_b))),V_c) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c) ) )).

fof(fact_zdiv__zmod__equality2,axiom,(
    ! [V_k,V_b,V_a] : hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b)),V_b)),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b))),V_k) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a),V_k) )).

fof(fact_zdiv__zmod__equality,axiom,(
    ! [V_k,V_a,V_b] : hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b))),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b))),V_k) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a),V_k) )).

fof(fact_zdiv__zmult1__eq,axiom,(
    ! [V_c,V_b,V_a] : hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),V_b)),V_c) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_b),V_c))),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_b,V_c))),V_c)) )).

fof(fact_zmod__zdiv__equality,axiom,(
    ! [V_b,V_a] : V_a = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b))),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b)) )).

fof(fact_zmod__zdiv__equality_H,axiom,(
    ! [V_n,V_m] : c_Divides_Odiv__class_Omod(tc_Int_Oint,V_m,V_n) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_m),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_m),V_n)),V_n)) )).

fof(fact_zmult__div__cancel,axiom,(
    ! [V_m,V_n] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_n),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_m),V_n)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_m),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_m,V_n)) )).

fof(fact_mod__self,axiom,(
    ! [V_a,T_a] :
      ( class_Divides_Osemiring__div(T_a)
     => c_Divides_Odiv__class_Omod(T_a,V_a,V_a) = c_Groups_Ozero__class_Ozero(T_a) ) )).

fof(fact_mod__by__0,axiom,(
    ! [V_a,T_a] :
      ( class_Divides_Osemiring__div(T_a)
     => c_Divides_Odiv__class_Omod(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a)) = V_a ) )).

fof(fact_mod__0,axiom,(
    ! [V_a,T_a] :
      ( class_Divides_Osemiring__div(T_a)
     => c_Divides_Odiv__class_Omod(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) = c_Groups_Ozero__class_Ozero(T_a) ) )).

fof(fact_mod__add__cong,axiom,(
    ! [V_b_H,V_b,V_a_H,V_c,V_a,T_a] :
      ( class_Divides_Osemiring__div(T_a)
     => ( c_Divides_Odiv__class_Omod(T_a,V_a,V_c) = c_Divides_Odiv__class_Omod(T_a,V_a_H,V_c)
       => ( c_Divides_Odiv__class_Omod(T_a,V_b,V_c) = c_Divides_Odiv__class_Omod(T_a,V_b_H,V_c)
         => c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a_H),V_b_H),V_c) ) ) ) )).

fof(fact_zmod__simps_I1_J,axiom,(
    ! [V_b,V_c,V_a,T_a] :
      ( class_Divides_Osemiring__div(T_a)
     => c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Divides_Odiv__class_Omod(T_a,V_a,V_c)),V_b),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b),V_c) ) )).

fof(fact_zmod__simps_I2_J,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Divides_Osemiring__div(T_a)
     => c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),c_Divides_Odiv__class_Omod(T_a,V_b,V_c)),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b),V_c) ) )).

fof(fact_mod__add__eq,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Divides_Osemiring__div(T_a)
     => c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Divides_Odiv__class_Omod(T_a,V_a,V_c)),c_Divides_Odiv__class_Omod(T_a,V_b,V_c)),V_c) ) )).

fof(fact_mod__add__left__eq,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Divides_Osemiring__div(T_a)
     => c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Divides_Odiv__class_Omod(T_a,V_a,V_c)),V_b),V_c) ) )).

fof(fact_mod__add__right__eq,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Divides_Osemiring__div(T_a)
     => c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),c_Divides_Odiv__class_Omod(T_a,V_b,V_c)),V_c) ) )).

fof(fact_mod__add__self1,axiom,(
    ! [V_a,V_b,T_a] :
      ( class_Divides_Osemiring__div(T_a)
     => c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_a),V_b) = c_Divides_Odiv__class_Omod(T_a,V_a,V_b) ) )).

fof(fact_mod__add__self2,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Divides_Osemiring__div(T_a)
     => c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b),V_b) = c_Divides_Odiv__class_Omod(T_a,V_a,V_b) ) )).

fof(fact_mod__mult__cong,axiom,(
    ! [V_b_H,V_b,V_a_H,V_c,V_a,T_a] :
      ( class_Divides_Osemiring__div(T_a)
     => ( c_Divides_Odiv__class_Omod(T_a,V_a,V_c) = c_Divides_Odiv__class_Omod(T_a,V_a_H,V_c)
       => ( c_Divides_Odiv__class_Omod(T_a,V_b,V_c) = c_Divides_Odiv__class_Omod(T_a,V_b_H,V_c)
         => c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_H),V_b_H),V_c) ) ) ) )).

fof(fact_zmod__simps_I4_J,axiom,(
    ! [V_b,V_c,V_a,T_a] :
      ( class_Divides_Osemiring__div(T_a)
     => c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Divides_Odiv__class_Omod(T_a,V_a,V_c)),V_b),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b),V_c) ) )).

fof(fact_mod__mult__mult2,axiom,(
    ! [V_b,V_c,V_a,T_a] :
      ( class_Divides_Osemiring__div(T_a)
     => c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Divides_Odiv__class_Omod(T_a,V_a,V_b)),V_c) ) )).

fof(fact_mod__mult__mult1,axiom,(
    ! [V_b,V_a,V_c,T_a] :
      ( class_Divides_Osemiring__div(T_a)
     => c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),c_Divides_Odiv__class_Omod(T_a,V_a,V_b)) ) )).

fof(fact_mod__mult__eq,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Divides_Osemiring__div(T_a)
     => c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Divides_Odiv__class_Omod(T_a,V_a,V_c)),c_Divides_Odiv__class_Omod(T_a,V_b,V_c)),V_c) ) )).

fof(fact_mod__mult__left__eq,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Divides_Osemiring__div(T_a)
     => c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Divides_Odiv__class_Omod(T_a,V_a,V_c)),V_b),V_c) ) )).

fof(fact_mod__mult__right__eq,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Divides_Osemiring__div(T_a)
     => c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Divides_Odiv__class_Omod(T_a,V_b,V_c)),V_c) ) )).

fof(fact_mod__diff__cong,axiom,(
    ! [V_b_H,V_b,V_a_H,V_c,V_a,T_a] :
      ( class_Divides_Oring__div(T_a)
     => ( c_Divides_Odiv__class_Omod(T_a,V_a,V_c) = c_Divides_Odiv__class_Omod(T_a,V_a_H,V_c)
       => ( c_Divides_Odiv__class_Omod(T_a,V_b,V_c) = c_Divides_Odiv__class_Omod(T_a,V_b_H,V_c)
         => c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_b),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a_H),V_b_H),V_c) ) ) ) )).

fof(fact_mod__diff__eq,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Divides_Oring__div(T_a)
     => c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_b),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),c_Divides_Odiv__class_Omod(T_a,V_a,V_c)),c_Divides_Odiv__class_Omod(T_a,V_b,V_c)),V_c) ) )).

fof(fact_mod__diff__left__eq,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Divides_Oring__div(T_a)
     => c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_b),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),c_Divides_Odiv__class_Omod(T_a,V_a,V_c)),V_b),V_c) ) )).

fof(fact_mod__diff__right__eq,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Divides_Oring__div(T_a)
     => c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_b),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),c_Divides_Odiv__class_Omod(T_a,V_b,V_c)),V_c) ) )).

fof(fact_mod__minus__eq,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Divides_Oring__div(T_a)
     => c_Divides_Odiv__class_Omod(T_a,hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a),V_b) = c_Divides_Odiv__class_Omod(T_a,hAPP(c_Groups_Ouminus__class_Ouminus(T_a),c_Divides_Odiv__class_Omod(T_a,V_a,V_b)),V_b) ) )).

fof(fact_mod__minus__cong,axiom,(
    ! [V_a_H,V_b,V_a,T_a] :
      ( class_Divides_Oring__div(T_a)
     => ( c_Divides_Odiv__class_Omod(T_a,V_a,V_b) = c_Divides_Odiv__class_Omod(T_a,V_a_H,V_b)
       => c_Divides_Odiv__class_Omod(T_a,hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a),V_b) = c_Divides_Odiv__class_Omod(T_a,hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a_H),V_b) ) ) )).

fof(fact_Compl__iff,axiom,(
    ! [V_A_2,V_c_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),V_A_2)))
    <=> ~ hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_A_2)) ) )).

fof(fact_ComplD,axiom,(
    ! [V_A_2,V_c_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),V_A_2)))
     => ~ hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_A_2)) ) )).

fof(fact_ComplE,axiom,(
    ! [V_A_2,V_c_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),V_A_2)))
     => ~ hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_A_2)) ) )).

fof(fact_zdiv__zero,axiom,(
    ! [V_b] : hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b) = c_Groups_Ozero__class_Ozero(tc_Int_Oint) )).

fof(fact_zmod__self,axiom,(
    ! [V_a] : c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_a) = c_Groups_Ozero__class_Ozero(tc_Int_Oint) )).

fof(fact_zmod__zero,axiom,(
    ! [V_b] : c_Divides_Odiv__class_Omod(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),V_b) = c_Groups_Ozero__class_Ozero(tc_Int_Oint) )).

fof(fact_Compl__subset__Compl__iff,axiom,(
    ! [V_B_2,V_A_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),V_A_2)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),V_B_2)))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2)) ) )).

fof(fact_Compl__anti__mono,axiom,(
    ! [V_B_2,V_A_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),V_B_2)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),V_A_2))) ) )).

fof(fact_zdiv__zminus2,axiom,(
    ! [V_b,V_a] : hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_b)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a)),V_b) )).

fof(fact_zdiv__zminus__zminus,axiom,(
    ! [V_b,V_a] : hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_b)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b) )).

fof(fact_zmod__zmult1__eq,axiom,(
    ! [V_c,V_b,V_a] : c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),V_b),V_c) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_b,V_c)),V_c) )).

fof(fact_zmod__simps_I3_J,axiom,(
    ! [V_c,V_b,V_a] : c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_b,V_c)),V_c) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),V_b),V_c) )).

fof(fact_zdiff__zmod__left,axiom,(
    ! [V_y,V_m,V_x] : c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_x,V_m)),V_y),V_m) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_x),V_y),V_m) )).

fof(fact_zdiff__zmod__right,axiom,(
    ! [V_m,V_y,V_x] : c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_x),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_y,V_m)),V_m) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_x),V_y),V_m) )).

fof(fact_zmod__zminus2,axiom,(
    ! [V_b,V_a] : c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_b)) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a),V_b)) )).

fof(fact_zmod__zminus__zminus,axiom,(
    ! [V_b,V_a] : c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_b)) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b)) )).

fof(fact_zminus__zmod,axiom,(
    ! [V_m,V_x] : c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_x,V_m)),V_m) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_x),V_m) )).

fof(fact_div__0,axiom,(
    ! [V_a,T_a] :
      ( class_Divides_Osemiring__div(T_a)
     => hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a) = c_Groups_Ozero__class_Ozero(T_a) ) )).

fof(fact_div__by__0,axiom,(
    ! [V_a,T_a] :
      ( class_Divides_Osemiring__div(T_a)
     => hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a)) = c_Groups_Ozero__class_Ozero(T_a) ) )).

fof(fact_div__by__1,axiom,(
    ! [V_a,T_a] :
      ( class_Divides_Osemiring__div(T_a)
     => hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_a),c_Groups_Oone__class_Oone(T_a)) = V_a ) )).

fof(fact_mod__mult__self1__is__0,axiom,(
    ! [V_a,V_b,T_a] :
      ( class_Divides_Osemiring__div(T_a)
     => c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a),V_b) = c_Groups_Ozero__class_Ozero(T_a) ) )).

fof(fact_mod__mult__self2__is__0,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Divides_Osemiring__div(T_a)
     => c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b),V_b) = c_Groups_Ozero__class_Ozero(T_a) ) )).

fof(fact_zmod__zmult2__eq,axiom,(
    ! [V_b,V_a,V_c] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_c))
     => c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),V_c)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b),V_c))),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b)) ) )).

fof(fact_mod__mult__self2,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Divides_Osemiring__div(T_a)
     => c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c)),V_b) = c_Divides_Odiv__class_Omod(T_a,V_a,V_b) ) )).

fof(fact_mod__mult__self1,axiom,(
    ! [V_b,V_c,V_a,T_a] :
      ( class_Divides_Osemiring__div(T_a)
     => c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b)),V_b) = c_Divides_Odiv__class_Omod(T_a,V_a,V_b) ) )).

fof(fact_mod__by__1,axiom,(
    ! [V_a,T_a] :
      ( class_Divides_Osemiring__div(T_a)
     => c_Divides_Odiv__class_Omod(T_a,V_a,c_Groups_Oone__class_Oone(T_a)) = c_Groups_Ozero__class_Ozero(T_a) ) )).

fof(fact_zdiv__zminus1__eq__if,axiom,(
    ! [V_a,V_b] :
      ( V_b != c_Groups_Ozero__class_Ozero(tc_Int_Oint)
     => ( ( c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b) = c_Groups_Ozero__class_Ozero(tc_Int_Oint)
         => hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a)),V_b) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b)) )
        & ( c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b) != c_Groups_Ozero__class_Ozero(tc_Int_Oint)
         => hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a)),V_b) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b))),c_Groups_Oone__class_Oone(tc_Int_Oint)) ) ) ) )).

fof(fact_zdiv__zminus2__eq__if,axiom,(
    ! [V_a,V_b] :
      ( V_b != c_Groups_Ozero__class_Ozero(tc_Int_Oint)
     => ( ( c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b) = c_Groups_Ozero__class_Ozero(tc_Int_Oint)
         => hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_b)) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b)) )
        & ( c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b) != c_Groups_Ozero__class_Ozero(tc_Int_Oint)
         => hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_b)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b))),c_Groups_Oone__class_Oone(tc_Int_Oint)) ) ) ) )).

fof(fact_iszero__def,axiom,(
    ! [V_z_2,T_b] :
      ( class_Rings_Osemiring__1(T_b)
     => ( c_Int_Oiszero(T_b,V_z_2)
      <=> V_z_2 = c_Groups_Ozero__class_Ozero(T_b) ) ) )).

fof(fact_iszero__0,axiom,(
    ! [T_a] :
      ( class_Rings_Osemiring__1(T_a)
     => c_Int_Oiszero(T_a,c_Groups_Ozero__class_Ozero(T_a)) ) )).

fof(fact_Divides_Otransfer__nat__int__function__closures_I1_J,axiom,(
    ! [V_y,V_x] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_y))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_x),V_y))) ) ) )).

fof(fact_not__iszero__1,axiom,(
    ! [T_a] :
      ( class_Rings_Osemiring__1(T_a)
     => ~ c_Int_Oiszero(T_a,c_Groups_Oone__class_Oone(T_a)) ) )).

fof(fact_div__neg__pos__less0,axiom,(
    ! [V_b,V_a] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_a),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) ) ) )).

fof(fact_neg__imp__zdiv__neg__iff,axiom,(
    ! [V_a_2,V_b_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a_2),V_b_2)),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a_2)) ) ) )).

fof(fact_pos__imp__zdiv__neg__iff,axiom,(
    ! [V_a_2,V_b_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a_2),V_b_2)),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_a_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) ) ) )).

fof(fact_zdiv__self,axiom,(
    ! [V_a] :
      ( V_a != c_Groups_Ozero__class_Ozero(tc_Int_Oint)
     => hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_a) = c_Groups_Oone__class_Oone(tc_Int_Oint) ) )).

fof(fact_subset__Compl__self__eq,axiom,(
    ! [V_A_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),V_A_2)))
    <=> V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) ) )).

fof(fact_Divides_Otransfer__nat__int__function__closures_I2_J,axiom,(
    ! [V_y,V_x] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_y))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_x,V_y))) ) ) )).

fof(fact_zmod__le__nonneg__dividend,axiom,(
    ! [V_k,V_m] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_m))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_m,V_k)),V_m)) ) )).

fof(fact_neg__mod__bound,axiom,(
    ! [V_a,V_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b))) ) )).

fof(fact_pos__mod__bound,axiom,(
    ! [V_a,V_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b)),V_b)) ) )).

fof(fact_zdiv__number__of__Bit0,axiom,(
    ! [V_w,V_v] : hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(V_v))),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(V_w))) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)) )).

fof(fact_zmod__eq__0__iff,axiom,(
    ! [V_d_2,V_m_2] :
      ( c_Divides_Odiv__class_Omod(tc_Int_Oint,V_m_2,V_d_2) = c_Groups_Ozero__class_Ozero(tc_Int_Oint)
    <=> ? [B_q] : V_m_2 = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_d_2),B_q) ) )).

fof(fact_zmod__zminus2__not__zero,axiom,(
    ! [V_l,V_k] :
      ( c_Divides_Odiv__class_Omod(tc_Int_Oint,V_k,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_l)) != c_Groups_Ozero__class_Ozero(tc_Int_Oint)
     => c_Divides_Odiv__class_Omod(tc_Int_Oint,V_k,V_l) != c_Groups_Ozero__class_Ozero(tc_Int_Oint) ) )).

fof(fact_zmod__zminus1__not__zero,axiom,(
    ! [V_l,V_k] :
      ( c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_k),V_l) != c_Groups_Ozero__class_Ozero(tc_Int_Oint)
     => c_Divides_Odiv__class_Omod(tc_Int_Oint,V_k,V_l) != c_Groups_Ozero__class_Ozero(tc_Int_Oint) ) )).

fof(fact_div__mult__mult1,axiom,(
    ! [V_b,V_a,V_c,T_a] :
      ( class_Divides_Osemiring__div(T_a)
     => ( V_c != c_Groups_Ozero__class_Ozero(T_a)
       => hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_a),V_b) ) ) )).

fof(fact_div__mult__mult2,axiom,(
    ! [V_b,V_a,V_c,T_a] :
      ( class_Divides_Osemiring__div(T_a)
     => ( V_c != c_Groups_Ozero__class_Ozero(T_a)
       => hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_a),V_b) ) ) )).

fof(fact_div__mult__self1__is__id,axiom,(
    ! [V_a,V_b,T_a] :
      ( class_Divides_Osemiring__div(T_a)
     => ( V_b != c_Groups_Ozero__class_Ozero(T_a)
       => hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a)),V_b) = V_a ) ) )).

fof(fact_div__mult__self2__is__id,axiom,(
    ! [V_a,V_b,T_a] :
      ( class_Divides_Osemiring__div(T_a)
     => ( V_b != c_Groups_Ozero__class_Ozero(T_a)
       => hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)),V_b) = V_a ) ) )).

fof(fact_div__mult__mult1__if,axiom,(
    ! [V_b,V_a,V_c,T_a] :
      ( class_Divides_Osemiring__div(T_a)
     => ( ( V_c = c_Groups_Ozero__class_Ozero(T_a)
         => hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b)) = c_Groups_Ozero__class_Ozero(T_a) )
        & ( V_c != c_Groups_Ozero__class_Ozero(T_a)
         => hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_a),V_b) ) ) ) )).

fof(fact_div__self,axiom,(
    ! [V_a,T_a] :
      ( class_Divides_Osemiring__div(T_a)
     => ( V_a != c_Groups_Ozero__class_Ozero(T_a)
       => hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_a),V_a) = c_Groups_Oone__class_Oone(T_a) ) ) )).

fof(fact_split__neg__lemma,axiom,(
    ! [V_n_2,V_P_2,V_k_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
     => ( hBOOL(hAPP(hAPP(V_P_2,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_n_2),V_k_2)),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_n_2,V_k_2)))
      <=> ! [B_i,B_j] :
            ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),B_j))
              & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B_j),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
              & V_n_2 = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_k_2),B_i)),B_j) )
           => hBOOL(hAPP(hAPP(V_P_2,B_i),B_j)) ) ) ) )).

fof(fact_split__pos__lemma,axiom,(
    ! [V_n_2,V_P_2,V_k_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_k_2))
     => ( hBOOL(hAPP(hAPP(V_P_2,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_n_2),V_k_2)),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_n_2,V_k_2)))
      <=> ! [B_i,B_j] :
            ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B_j))
              & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B_j),V_k_2))
              & V_n_2 = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_k_2),B_i)),B_j) )
           => hBOOL(hAPP(hAPP(V_P_2,B_i),B_j)) ) ) ) )).

fof(fact_zdiv__mono1__neg,axiom,(
    ! [V_b,V_a_H,V_a] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_a),V_a_H))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a_H),V_b)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b))) ) ) )).

fof(fact_zdiv__mono1,axiom,(
    ! [V_b,V_a_H,V_a] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_a),V_a_H))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a_H),V_b))) ) ) )).

fof(fact_div__neg__neg__trivial,axiom,(
    ! [V_b,V_a] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_a),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b),V_a))
       => hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b) = c_Groups_Ozero__class_Ozero(tc_Int_Oint) ) ) )).

fof(fact_zdiv__mono2__neg,axiom,(
    ! [V_b,V_b_H,V_a] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_a),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b_H))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_b_H),V_b))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b_H)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b))) ) ) ) )).

fof(fact_div__nonpos__pos__le0,axiom,(
    ! [V_b,V_a] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_a),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) ) ) )).

fof(fact_neg__imp__zdiv__nonneg__iff,axiom,(
    ! [V_a_2,V_b_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a_2),V_b_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_a_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) ) ) )).

fof(fact_div__pos__pos__trivial,axiom,(
    ! [V_b,V_a] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_a),V_b))
       => hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b) = c_Groups_Ozero__class_Ozero(tc_Int_Oint) ) ) )).

fof(fact_div__nonneg__neg__le0,axiom,(
    ! [V_b,V_a] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) ) ) )).

fof(fact_zdiv__mono2,axiom,(
    ! [V_b,V_b_H,V_a] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b_H))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_b_H),V_b))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b_H))) ) ) ) )).

fof(fact_nonneg1__imp__zdiv__pos__iff,axiom,(
    ! [V_b_2,V_a_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a_2),V_b_2)))
      <=> ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_b_2),V_a_2))
          & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b_2)) ) ) ) )).

fof(fact_pos__imp__zdiv__pos__iff,axiom,(
    ! [V_i_2,V_k_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_k_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_i_2),V_k_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k_2),V_i_2)) ) ) )).

fof(fact_pos__imp__zdiv__nonneg__iff,axiom,(
    ! [V_a_2,V_b_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a_2),V_b_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a_2)) ) ) )).

fof(fact_zdiv__eq__0__iff,axiom,(
    ! [V_k_2,V_i_2] :
      ( hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_i_2),V_k_2) = c_Groups_Ozero__class_Ozero(tc_Int_Oint)
    <=> ( V_k_2 = c_Groups_Ozero__class_Ozero(tc_Int_Oint)
        | ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_i_2))
          & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_i_2),V_k_2)) )
        | ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_i_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
          & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),V_i_2)) ) ) ) )).

fof(fact_int__div__less__self,axiom,(
    ! [V_k,V_x] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),V_k))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_x),V_k)),V_x)) ) ) )).

fof(fact_zdiv__zmult2__eq,axiom,(
    ! [V_b,V_a,V_c] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_c))
     => hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),V_c)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b)),V_c) ) )).

fof(fact_mod__neg__neg__trivial,axiom,(
    ! [V_b,V_a] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_a),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b),V_a))
       => c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b) = V_a ) ) )).

fof(fact_neg__mod__conj,axiom,(
    ! [V_a,V_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b)),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
        & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b))) ) ) )).

fof(fact_neg__mod__sign,axiom,(
    ! [V_a,V_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) ) )).

fof(fact_mod__pos__pos__trivial,axiom,(
    ! [V_b,V_a] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_a),V_b))
       => c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b) = V_a ) ) )).

fof(fact_pos__mod__conj,axiom,(
    ! [V_a,V_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b)))
        & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b)),V_b)) ) ) )).

fof(fact_pos__mod__sign,axiom,(
    ! [V_a,V_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b))) ) )).

fof(fact_zmod__zminus1__eq__if,axiom,(
    ! [V_b,V_a] :
      ( ( c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b) = c_Groups_Ozero__class_Ozero(tc_Int_Oint)
       => c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a),V_b) = c_Groups_Ozero__class_Ozero(tc_Int_Oint) )
      & ( c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b) != c_Groups_Ozero__class_Ozero(tc_Int_Oint)
       => c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a),V_b) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_b),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b)) ) ) )).

fof(fact_zmod__zminus2__eq__if,axiom,(
    ! [V_b,V_a] :
      ( ( c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b) = c_Groups_Ozero__class_Ozero(tc_Int_Oint)
       => c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_b)) = c_Groups_Ozero__class_Ozero(tc_Int_Oint) )
      & ( c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b) != c_Groups_Ozero__class_Ozero(tc_Int_Oint)
       => c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_b)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b)),V_b) ) ) )).

fof(fact_div__mult__self2,axiom,(
    ! [V_c,V_a,V_b,T_a] :
      ( class_Divides_Osemiring__div(T_a)
     => ( V_b != c_Groups_Ozero__class_Ozero(T_a)
       => hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c))),V_b) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_a),V_b)) ) ) )).

fof(fact_div__mult__self1,axiom,(
    ! [V_c,V_a,V_b,T_a] :
      ( class_Divides_Osemiring__div(T_a)
     => ( V_b != c_Groups_Ozero__class_Ozero(T_a)
       => hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b))),V_b) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_a),V_b)) ) ) )).

fof(fact_div__add__self2,axiom,(
    ! [V_a,V_b,T_a] :
      ( class_Divides_Osemiring__div(T_a)
     => ( V_b != c_Groups_Ozero__class_Ozero(T_a)
       => hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),V_b) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_a),V_b)),c_Groups_Oone__class_Oone(T_a)) ) ) )).

fof(fact_div__add__self1,axiom,(
    ! [V_a,V_b,T_a] :
      ( class_Divides_Osemiring__div(T_a)
     => ( V_b != c_Groups_Ozero__class_Ozero(T_a)
       => hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_a)),V_b) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_a),V_b)),c_Groups_Oone__class_Oone(T_a)) ) ) )).

fof(fact_iszero__Numeral0,axiom,(
    ! [T_a] :
      ( class_Int_Onumber__ring(T_a)
     => c_Int_Oiszero(T_a,hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OPls)) ) )).

fof(fact_iszero__number__of__Bit1,axiom,(
    ! [V_w,T_a] :
      ( ( class_Int_Onumber__ring(T_a)
        & class_Int_Oring__char__0(T_a) )
     => ~ c_Int_Oiszero(T_a,hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit1(V_w))) ) )).

fof(fact_iszero__number__of__Bit0,axiom,(
    ! [V_w_2,T_b] :
      ( ( class_Int_Onumber__ring(T_b)
        & class_Int_Oring__char__0(T_b) )
     => ( c_Int_Oiszero(T_b,hAPP(c_Int_Onumber__class_Onumber__of(T_b),c_Int_OBit0(V_w_2)))
      <=> c_Int_Oiszero(T_b,hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)) ) ) )).

fof(fact_mod__pos__neg__trivial,axiom,(
    ! [V_b,V_a] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a),V_b)),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
       => c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a),V_b) ) ) )).

fof(fact_divmod__int__rel__div__eq,axiom,(
    ! [V_r_1,V_y,V_b_1,V_a_1] :
      ( V_a_1 = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b_1),V_y)),V_r_1)
     => ( ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b_1))
           => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_r_1))
              & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_r_1),V_b_1)) ) )
          & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b_1))
           => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b_1),V_r_1))
              & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_r_1),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) ) ) )
       => ( V_b_1 != c_Groups_Ozero__class_Ozero(tc_Int_Oint)
         => hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a_1),V_b_1) = V_y ) ) ) )).

fof(fact_split__zdiv,axiom,(
    ! [V_k_2,V_n_2,V_P_2] :
      ( hBOOL(hAPP(V_P_2,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_n_2),V_k_2)))
    <=> ( ( V_k_2 = c_Groups_Ozero__class_Ozero(tc_Int_Oint)
         => hBOOL(hAPP(V_P_2,c_Groups_Ozero__class_Ozero(tc_Int_Oint))) )
        & ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_k_2))
         => ! [B_i] :
              ( ? [B_j] :
                  ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B_j))
                  & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B_j),V_k_2))
                  & V_n_2 = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_k_2),B_i)),B_j) )
             => hBOOL(hAPP(V_P_2,B_i)) ) )
        & ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
         => ! [B_i] :
              ( ? [B_j] :
                  ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),B_j))
                  & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B_j),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
                  & V_n_2 = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_k_2),B_i)),B_j) )
             => hBOOL(hAPP(V_P_2,B_i)) ) ) ) ) )).

fof(fact_divmod__int__rel__mod__eq,axiom,(
    ! [V_y,V_q_1,V_b_1,V_a_1] :
      ( V_a_1 = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b_1),V_q_1)),V_y)
     => ( ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b_1))
           => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_y))
              & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_y),V_b_1)) ) )
          & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b_1))
           => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b_1),V_y))
              & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_y),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) ) ) )
       => ( V_b_1 != c_Groups_Ozero__class_Ozero(tc_Int_Oint)
         => c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a_1,V_b_1) = V_y ) ) ) )).

fof(fact_zmult2__lemma__aux2,axiom,(
    ! [V_q,V_r,V_b,V_c] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_c))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b),V_r))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_r),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_q,V_c))),V_r)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) ) ) ) )).

fof(fact_zmult2__lemma__aux1,axiom,(
    ! [V_q,V_r,V_b,V_c] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_c))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b),V_r))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_r),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),V_c)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_q,V_c))),V_r))) ) ) ) )).

fof(fact_zmult2__lemma__aux4,axiom,(
    ! [V_q,V_b,V_r,V_c] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_c))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_r))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_r),V_b))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_q,V_c))),V_r)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),V_c))) ) ) ) )).

fof(fact_zmult2__lemma__aux3,axiom,(
    ! [V_q,V_b,V_r,V_c] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_c))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_r))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_r),V_b))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_q,V_c))),V_r))) ) ) ) )).

fof(fact_split__zmod,axiom,(
    ! [V_k_2,V_n_2,V_P_2] :
      ( hBOOL(hAPP(V_P_2,c_Divides_Odiv__class_Omod(tc_Int_Oint,V_n_2,V_k_2)))
    <=> ( ( V_k_2 = c_Groups_Ozero__class_Ozero(tc_Int_Oint)
         => hBOOL(hAPP(V_P_2,V_n_2)) )
        & ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_k_2))
         => ! [B_i,B_j] :
              ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B_j))
                & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B_j),V_k_2))
                & V_n_2 = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_k_2),B_i)),B_j) )
             => hBOOL(hAPP(V_P_2,B_j)) ) )
        & ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
         => ! [B_i,B_j] :
              ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),B_j))
                & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B_j),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
                & V_n_2 = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_k_2),B_i)),B_j) )
             => hBOOL(hAPP(V_P_2,B_j)) ) ) ) ) )).

fof(fact_zmod__number__of__Bit0,axiom,(
    ! [V_w,V_v] : c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(V_w))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w))) )).

fof(fact_not__iszero__Numeral1,axiom,(
    ! [T_a] :
      ( class_Int_Onumber__ring(T_a)
     => ~ c_Int_Oiszero(T_a,hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit1(c_Int_OPls))) ) )).

fof(fact_eq__number__of__eq,axiom,(
    ! [V_y_2,V_x_2,T_b] :
      ( class_Int_Onumber__ring(T_b)
     => ( hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_x_2) = hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_y_2)
      <=> c_Int_Oiszero(T_b,hAPP(c_Int_Onumber__class_Onumber__of(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_x_2),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_y_2)))) ) ) )).

fof(fact_zdiv__number__of__Bit1,axiom,(
    ! [V_v,V_w] :
      ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)))
       => hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit1(V_v))),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(V_w))) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)) )
      & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)))
       => hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit1(V_v))),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(V_w))) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)),c_Groups_Oone__class_Oone(tc_Int_Oint))),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)) ) ) )).

fof(fact_eq__special_I1_J,axiom,(
    ! [V_y_2,T_b] :
      ( class_Int_Onumber__ring(T_b)
     => ( c_Groups_Ozero__class_Ozero(T_b) = hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_y_2)
      <=> c_Int_Oiszero(T_b,hAPP(c_Int_Onumber__class_Onumber__of(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Int_OPls),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_y_2)))) ) ) )).

fof(fact_eq__special_I3_J,axiom,(
    ! [V_x_2,T_b] :
      ( class_Int_Onumber__ring(T_b)
     => ( hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_x_2) = c_Groups_Ozero__class_Ozero(T_b)
      <=> c_Int_Oiszero(T_b,hAPP(c_Int_Onumber__class_Onumber__of(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_x_2),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Int_OPls)))) ) ) )).

fof(fact_nat__of__aux__code,axiom,(
    ! [V_n,V_i] :
      ( ( V_i = c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)
       => c_Code__Numeral_Onat__of__aux(V_i,V_n) = V_n )
      & ( V_i != c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)
       => c_Code__Numeral_Onat__of__aux(V_i,V_n) = c_Code__Numeral_Onat__of__aux(hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Code__Numeral_Ocode__numeral),V_i),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),hAPP(c_Nat_OSuc,V_n)) ) ) )).

fof(fact_z3div__def,axiom,(
    ! [V_k,V_l] :
      ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_l))
       => c_SMT_Oz3div(V_k,V_l) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_k),V_l) )
      & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_l))
       => c_SMT_Oz3div(V_k,V_l) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_k),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_l))) ) ) )).

fof(fact_z3mod__def,axiom,(
    ! [V_k,V_l] :
      ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_l))
       => c_SMT_Oz3mod(V_k,V_l) = c_Divides_Odiv__class_Omod(tc_Int_Oint,V_k,V_l) )
      & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_l))
       => c_SMT_Oz3mod(V_k,V_l) = c_Divides_Odiv__class_Omod(tc_Int_Oint,V_k,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_l)) ) ) )).

fof(fact_of__nat__double,axiom,(
    ! [V_x,T_a] :
      ( class_Rings_Osemiring__1(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x))
       => hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),hAPP(c_Int_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_x))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),hAPP(c_Int_Onat,V_x))),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),hAPP(c_Int_Onat,V_x))) ) ) )).

fof(fact_power__minus1__odd,axiom,(
    ! [V_n,T_a] :
      ( class_Rings_Oring__1(T_a)
     => hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),c_Groups_Oone__class_Oone(T_a))),hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_n))) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),c_Groups_Oone__class_Oone(T_a)) ) )).

fof(fact_odd__power__less__zero,axiom,(
    ! [V_n,V_a,T_a] :
      ( class_Rings_Olinordered__idom(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a)))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_n)))),c_Groups_Ozero__class_Ozero(T_a))) ) ) )).

fof(fact_Compl__eq__Compl__iff,axiom,(
    ! [V_B_2,V_A_2,T_b] :
      ( hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),V_A_2) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),V_B_2)
    <=> V_A_2 = V_B_2 ) )).

fof(fact_double__complement,axiom,(
    ! [V_A_2,T_b] : hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),V_A_2)) = V_A_2 )).

fof(fact_Nat__Transfer_Otransfer__nat__int__functions_I4_J,axiom,(
    ! [V_n,V_x] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x))
     => hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),hAPP(c_Int_Onat,V_x)),V_n) = hAPP(c_Int_Onat,hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_x),V_n)) ) )).

fof(fact_div__add1__eq,axiom,(
    ! [V_c,V_b,V_a] : hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_a),V_b)),V_c) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_a),V_c)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_b),V_c))),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_a,V_c)),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_b,V_c))),V_c)) )).

fof(fact_nat__power__eq,axiom,(
    ! [V_n,V_z] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z))
     => hAPP(c_Int_Onat,hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_z),V_n)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),hAPP(c_Int_Onat,V_z)),V_n) ) )).

fof(fact_div__mult1__eq,axiom,(
    ! [V_c,V_b,V_a] : hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_a),V_b)),V_c) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_a),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_b),V_c))),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_a),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_b,V_c))),V_c)) )).

fof(fact_mod__mult2__eq,axiom,(
    ! [V_c,V_b,V_a] : c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_b),V_c)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_b),c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_a),V_b),V_c))),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_a,V_b)) )).

fof(fact_zpower__int,axiom,(
    ! [V_n,V_m] : hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m)),V_n) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_m),V_n)) )).

fof(fact_int__power,axiom,(
    ! [V_n,V_m] : hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_m),V_n)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m)),V_n) )).

fof(fact_Nat__Transfer_Otransfer__int__nat__functions_I4_J,axiom,(
    ! [V_n,V_x] : hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_x)),V_n) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_x),V_n)) )).

fof(fact_div__mod__equality_H,axiom,(
    ! [V_n,V_m] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),V_n)),V_n) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n)) )).

fof(fact_mult__div__cancel,axiom,(
    ! [V_m,V_n] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),V_n)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n)) )).

fof(fact_Divides_Omod__div__equality_H,axiom,(
    ! [V_n,V_m] : c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),V_n)),V_n)) )).

fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,(
    ! [V_q,V_y,V_x,T_a] :
      ( class_Rings_Ocomm__semiring__1(T_a)
     => hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_y)),V_q) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_q)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_y),V_q)) ) )).

fof(fact_div__le__dividend,axiom,(
    ! [V_n,V_m] : hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),V_n)),V_m)) )).

fof(fact_div__le__mono,axiom,(
    ! [V_k,V_n,V_m] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),V_n))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),V_k)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_n),V_k))) ) )).

fof(fact_mod__Suc__eq__Suc__mod,axiom,(
    ! [V_n,V_m] : c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m),V_n) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Nat_OSuc,c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n)),V_n) )).

fof(fact_div__mult2__eq,axiom,(
    ! [V_c,V_b,V_a] : hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_b),V_c)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_a),V_b)),V_c) )).

fof(fact_mod__less,axiom,(
    ! [V_n,V_m] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n))
     => c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n) = V_m ) )).

fof(fact_mod__less__eq__dividend,axiom,(
    ! [V_n,V_m] : hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n)),V_m)) )).

fof(fact_mod__mult__distrib,axiom,(
    ! [V_k,V_n,V_m] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n)),V_k) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_k),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n),V_k)) )).

fof(fact_mod__mult__distrib2,axiom,(
    ! [V_n,V_m,V_k] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n)) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_m),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_n)) )).

fof(fact_nat__int,axiom,(
    ! [V_n] : hAPP(c_Int_Onat,hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n)) = V_n )).

fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,(
    ! [V_q,V_p,V_x,T_a] :
      ( class_Rings_Ocomm__semiring__1(T_a)
     => hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_p)),V_q) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_p),V_q)) ) )).

fof(fact_zpower__zpower,axiom,(
    ! [V_z,V_y,V_x] : hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_x),V_y)),V_z) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_y),V_z)) )).

fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,(
    ! [V_x,T_a] :
      ( class_Rings_Ocomm__semiring__1(T_a)
     => hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = V_x ) )).

fof(fact_zpower__zmod,axiom,(
    ! [V_y,V_m,V_x] : c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_x,V_m)),V_y),V_m) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_x),V_y),V_m) )).

fof(fact_power__nat__number__of__number__of,axiom,(
    ! [V_w,V_v] :
      ( ( hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)))
       => hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_w)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_w)) )
      & ( ~ hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)))
       => hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_w)) = hAPP(c_Int_Onat,hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_w))) ) ) )).

fof(fact_power__nat__number__of,axiom,(
    ! [V_n,V_v] :
      ( ( hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)))
       => hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),V_n) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n) )
      & ( ~ hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)))
       => hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),V_n) = hAPP(c_Int_Onat,hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)),V_n)) ) ) )).

fof(fact_Divides_Otransfer__nat__int__functions_I1_J,axiom,(
    ! [V_y,V_x] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_y))
       => hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(c_Int_Onat,V_x)),hAPP(c_Int_Onat,V_y)) = hAPP(c_Int_Onat,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_x),V_y)) ) ) )).

fof(fact_nat__div__distrib,axiom,(
    ! [V_y,V_x] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x))
     => hAPP(c_Int_Onat,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_x),V_y)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(c_Int_Onat,V_x)),hAPP(c_Int_Onat,V_y)) ) )).

fof(fact_Divides_Otransfer__nat__int__functions_I2_J,axiom,(
    ! [V_y,V_x] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_y))
       => c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Int_Onat,V_x),hAPP(c_Int_Onat,V_y)) = hAPP(c_Int_Onat,c_Divides_Odiv__class_Omod(tc_Int_Oint,V_x,V_y)) ) ) )).

fof(fact_nat__mod__distrib,axiom,(
    ! [V_y,V_x] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_y))
       => hAPP(c_Int_Onat,c_Divides_Odiv__class_Omod(tc_Int_Oint,V_x,V_y)) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Int_Onat,V_x),hAPP(c_Int_Onat,V_y)) ) ) )).

fof(fact_div__1,axiom,(
    ! [V_m] : hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) = V_m )).

fof(fact_div__less,axiom,(
    ! [V_n,V_m] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n))
     => hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),V_n) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )).

fof(fact_nat__mult__div__cancel__disj,axiom,(
    ! [V_n,V_m,V_k] :
      ( ( V_k = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
       => hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_m)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_n)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )
      & ( V_k != c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
       => hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_m)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_n)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),V_n) ) ) )).

fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J,axiom,(
    ! [V_q,V_x,T_a] :
      ( class_Rings_Ocomm__semiring__1(T_a)
     => hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),hAPP(c_Nat_OSuc,V_q)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_q)) ) )).

fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J,axiom,(
    ! [V_q,V_x,T_a] :
      ( class_Rings_Ocomm__semiring__1(T_a)
     => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_q)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),hAPP(c_Nat_OSuc,V_q)) ) )).

fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J,axiom,(
    ! [V_q,V_x,T_a] :
      ( class_Rings_Ocomm__semiring__1(T_a)
     => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_q)),V_x) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),hAPP(c_Nat_OSuc,V_q)) ) )).

fof(fact_transfer__nat__int__numerals_I1_J,axiom,(
    c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(c_Int_Onat,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) )).

fof(fact_nat__0,axiom,(
    hAPP(c_Int_Onat,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )).

fof(fact_mod__1,axiom,(
    ! [V_m] : c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )).

fof(fact_mod__Suc,axiom,(
    ! [V_n,V_m] :
      ( ( hAPP(c_Nat_OSuc,c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n)) = V_n
       => c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m),V_n) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )
      & ( hAPP(c_Nat_OSuc,c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n)) != V_n
       => c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m),V_n) = hAPP(c_Nat_OSuc,c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n)) ) ) )).

fof(fact_mod__less__divisor,axiom,(
    ! [V_m,V_n] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n)),V_n)) ) )).

fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,(
    ! [V_x,T_a] :
      ( class_Rings_Ocomm__semiring__1(T_a)
     => hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = c_Groups_Oone__class_Oone(T_a) ) )).

fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,(
    ! [V_q,V_p,V_x,T_a] :
      ( class_Rings_Ocomm__semiring__1(T_a)
     => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_p)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_q)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_p),V_q)) ) )).

fof(fact_ex__nat,axiom,(
    ! [V_P_2] :
      ( ? [B_x1] : hBOOL(hAPP(V_P_2,B_x1))
    <=> ? [B_x] :
          ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B_x))
          & hBOOL(hAPP(V_P_2,hAPP(c_Int_Onat,B_x))) ) ) )).

fof(fact_all__nat,axiom,(
    ! [V_P_2] :
      ( ! [B_x1] : hBOOL(hAPP(V_P_2,B_x1))
    <=> ! [B_x] :
          ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B_x))
         => hBOOL(hAPP(V_P_2,hAPP(c_Int_Onat,B_x))) ) ) )).

fof(fact_transfer__nat__int__relations_I1_J,axiom,(
    ! [V_y_2,V_x_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_y_2))
       => ( hAPP(c_Int_Onat,V_x_2) = hAPP(c_Int_Onat,V_y_2)
        <=> V_x_2 = V_y_2 ) ) ) )).

fof(fact_eq__nat__nat__iff,axiom,(
    ! [V_z_H_2,V_z_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z_H_2))
       => ( hAPP(c_Int_Onat,V_z_2) = hAPP(c_Int_Onat,V_z_H_2)
        <=> V_z_2 = V_z_H_2 ) ) ) )).

fof(fact_mod__eq__0__iff,axiom,(
    ! [V_d_2,V_m_2] :
      ( c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m_2,V_d_2) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
    <=> ? [B_q] : V_m_2 = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_d_2),B_q) ) )).

fof(fact_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J,axiom,(
    ! [V_n,V_x] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_x),V_n))) ) )).

fof(fact_mod__if,axiom,(
    ! [V_n,V_m] :
      ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n))
       => c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n) = V_m )
      & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n))
       => c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n),V_n) ) ) )).

fof(fact_mod__geq,axiom,(
    ! [V_n,V_m] :
      ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n))
     => c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n),V_n) ) )).

fof(fact_mod__mult__self3,axiom,(
    ! [V_m,V_n,V_k] : c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_n)),V_m),V_n) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n) )).

fof(fact_le__mod__geq,axiom,(
    ! [V_m,V_n] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n),V_m))
     => c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n),V_n) ) )).

fof(fact_Divides_Otransfer__int__nat__functions_I1_J,axiom,(
    ! [V_y,V_x] : hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_x)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_y)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_x),V_y)) )).

fof(fact_zdiv__int,axiom,(
    ! [V_b,V_a] : hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_a),V_b)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_a)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_b)) )).

fof(fact_nat__number__of__def,axiom,(
    ! [V_v] : hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v) = hAPP(c_Int_Onat,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)) )).

fof(fact_nat__number__of,axiom,(
    ! [V_w] : hAPP(c_Int_Onat,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_w) )).

fof(fact_zpower__zadd__distrib,axiom,(
    ! [V_z,V_y,V_x] : hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_x),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_y),V_z)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_x),V_y)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_x),V_z)) )).

fof(fact_transfer__nat__int__numerals_I2_J,axiom,(
    c_Groups_Oone__class_Oone(tc_Nat_Onat) = hAPP(c_Int_Onat,c_Groups_Oone__class_Oone(tc_Int_Oint)) )).

fof(fact_Divides_Otransfer__int__nat__functions_I2_J,axiom,(
    ! [V_y,V_x] : c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_x),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_y)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_x,V_y)) )).

fof(fact_zmod__int,axiom,(
    ! [V_b,V_a] : hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_a,V_b)) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_a),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_b)) )).

fof(fact_neg__nat,axiom,(
    ! [V_z] :
      ( hBOOL(hAPP(c_Nat__Numeral_Oneg,V_z))
     => hAPP(c_Int_Onat,V_z) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )).

fof(fact_not__neg__nat,axiom,(
    ! [V_z] :
      ( ~ hBOOL(hAPP(c_Nat__Numeral_Oneg,V_z))
     => hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Int_Onat,V_z)) = V_z ) )).

fof(fact_Ints__power,axiom,(
    ! [V_n_2,V_a_2,T_b] :
      ( class_Rings_Oring__1(T_b)
     => ( hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),c_Int_Oring__1__class_OInts(T_b)))
       => hBOOL(hAPP(hAPP(c_member(T_b),hAPP(hAPP(c_Power_Opower__class_Opower(T_b),V_a_2),V_n_2)),c_Int_Oring__1__class_OInts(T_b))) ) ) )).

fof(fact_div__nat__number__of,axiom,(
    ! [V_v_H,V_v] :
      ( ( hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)))
       => hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )
      & ( ~ hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)))
       => hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H)) = hAPP(c_Int_Onat,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_H))) ) ) )).

fof(fact_mod__nat__number__of,axiom,(
    ! [V_v_H,V_v] :
      ( ( hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)))
       => c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )
      & ( ~ hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)))
       => ( ( hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_H)))
           => c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H)) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v) )
          & ( ~ hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_H)))
           => c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H)) = hAPP(c_Int_Onat,c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_H))) ) ) ) ) )).

fof(fact_one__div__nat__number__of,axiom,(
    ! [V_v_H] : hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H)) = hAPP(c_Int_Onat,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_H))) )).

fof(fact_div__le__mono2,axiom,(
    ! [V_k,V_n,V_m] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_m))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),V_n))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_k),V_n)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_k),V_m))) ) ) )).

fof(fact_nat__mult__div__cancel1,axiom,(
    ! [V_n,V_m,V_k] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_k))
     => hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_m)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_n)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),V_n) ) )).

fof(fact_div__mult__self__is__m,axiom,(
    ! [V_m,V_n] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n))
     => hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n)),V_n) = V_m ) )).

fof(fact_div__mult__self1__is__m,axiom,(
    ! [V_m,V_n] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n))
     => hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n),V_m)),V_n) = V_m ) )).

fof(fact_div__less__dividend,axiom,(
    ! [V_m,V_n] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),V_n))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_m))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),V_n)),V_m)) ) ) )).

fof(fact_nat__0__iff,axiom,(
    ! [V_i_2] :
      ( hAPP(c_Int_Onat,V_i_2) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_i_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) ) )).

fof(fact_nat__le__0,axiom,(
    ! [V_z] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_z),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
     => hAPP(c_Int_Onat,V_z) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) )).

fof(fact_zless__nat__conj,axiom,(
    ! [V_z_2,V_w_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onat,V_w_2)),hAPP(c_Int_Onat,V_z_2)))
    <=> ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z_2))
        & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_w_2),V_z_2)) ) ) )).

fof(fact_nat__mono__iff,axiom,(
    ! [V_w_2,V_z_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onat,V_w_2)),hAPP(c_Int_Onat,V_z_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_w_2),V_z_2)) ) ) )).

fof(fact_mod__le__divisor,axiom,(
    ! [V_m,V_n] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n)),V_n)) ) )).

fof(fact_transfer__nat__int__relations_I3_J,axiom,(
    ! [V_y_2,V_x_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_y_2))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Int_Onat,V_x_2)),hAPP(c_Int_Onat,V_y_2)))
        <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_x_2),V_y_2)) ) ) ) )).

fof(fact_nat__1,axiom,(
    hAPP(c_Int_Onat,c_Groups_Oone__class_Oone(tc_Int_Oint)) = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) )).

fof(fact_mod__mult__self4,axiom,(
    ! [V_m,V_n,V_k] : c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_n)),V_m)),V_n) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m),V_n) )).

fof(fact_int__nat__eq,axiom,(
    ! [V_z] :
      ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z))
       => hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Int_Onat,V_z)) = V_z )
      & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z))
       => hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Int_Onat,V_z)) = c_Groups_Ozero__class_Ozero(tc_Int_Oint) ) ) )).

fof(fact_int__eq__iff,axiom,(
    ! [V_z_2,V_m_2] :
      ( hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m_2) = V_z_2
    <=> ( V_m_2 = hAPP(c_Int_Onat,V_z_2)
        & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z_2)) ) ) )).

fof(fact_nat__0__le,axiom,(
    ! [V_z] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z))
     => hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Int_Onat,V_z)) = V_z ) )).

fof(fact_zless__nat__eq__int__zless,axiom,(
    ! [V_z_2,V_m_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),hAPP(c_Int_Onat,V_z_2)))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m_2)),V_z_2)) ) )).

fof(fact_nat__zminus__int,axiom,(
    ! [V_n] : hAPP(c_Int_Onat,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n))) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )).

fof(fact_zero__le__zpower__abs,axiom,(
    ! [V_n,V_x] : hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_x)),V_n))) )).

fof(fact_nat__abs__mult__distrib,axiom,(
    ! [V_z,V_w] : hAPP(c_Int_Onat,hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_w),V_z))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onat,hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_w))),hAPP(c_Int_Onat,hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_z))) )).

fof(fact_one__mod__nat__number__of,axiom,(
    ! [V_v_H] :
      ( ( hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_H)))
       => c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H)) = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) )
      & ( ~ hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_H)))
       => c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H)) = hAPP(c_Int_Onat,c_Divides_Odiv__class_Omod(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_H))) ) ) )).

fof(fact_div__geq,axiom,(
    ! [V_m,V_n] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n))
     => ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n))
       => hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),V_n) = hAPP(c_Nat_OSuc,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)),V_n)) ) ) )).

fof(fact_div__if,axiom,(
    ! [V_m,V_n] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n))
     => ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n))
         => hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),V_n) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )
        & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n))
         => hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),V_n) = hAPP(c_Nat_OSuc,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)),V_n)) ) ) ) )).

fof(fact_split__div,axiom,(
    ! [V_k_2,V_n_2,V_P_2] :
      ( hBOOL(hAPP(V_P_2,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_n_2),V_k_2)))
    <=> ( ( V_k_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
         => hBOOL(hAPP(V_P_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) )
        & ( V_k_2 != c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
         => ! [B_i,B_j] :
              ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_j),V_k_2))
             => ( V_n_2 = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),B_i)),B_j)
               => hBOOL(hAPP(V_P_2,B_i)) ) ) ) ) ) )).

fof(fact_zero__less__nat__eq,axiom,(
    ! [V_z_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Int_Onat,V_z_2)))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z_2)) ) )).

fof(fact_transfer__nat__int__relations_I2_J,axiom,(
    ! [V_y_2,V_x_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_y_2))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onat,V_x_2)),hAPP(c_Int_Onat,V_y_2)))
        <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_x_2),V_y_2)) ) ) ) )).

fof(fact_nat__less__eq__zless,axiom,(
    ! [V_z_2,V_w_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_w_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onat,V_w_2)),hAPP(c_Int_Onat,V_z_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_w_2),V_z_2)) ) ) )).

fof(fact_nat__eq__iff,axiom,(
    ! [V_m_2,V_w_2] :
      ( hAPP(c_Int_Onat,V_w_2) = V_m_2
    <=> ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_w_2))
         => V_w_2 = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m_2) )
        & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_w_2))
         => V_m_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) ) ) )).

fof(fact_nat__eq__iff2,axiom,(
    ! [V_w_2,V_m_2] :
      ( V_m_2 = hAPP(c_Int_Onat,V_w_2)
    <=> ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_w_2))
         => V_w_2 = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m_2) )
        & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_w_2))
         => V_m_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) ) ) )).

fof(fact_nat__le__eq__zle,axiom,(
    ! [V_z_2,V_w_2] :
      ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_w_2))
        | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z_2)) )
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Int_Onat,V_w_2)),hAPP(c_Int_Onat,V_z_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_w_2),V_z_2)) ) ) )).

fof(fact_split__nat,axiom,(
    ! [V_i_2,V_P_2] :
      ( hBOOL(hAPP(V_P_2,hAPP(c_Int_Onat,V_i_2)))
    <=> ( ! [B_n] :
            ( V_i_2 = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),B_n)
           => hBOOL(hAPP(V_P_2,B_n)) )
        & ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_i_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
         => hBOOL(hAPP(V_P_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) ) ) ) )).

fof(fact_split__mod,axiom,(
    ! [V_k_2,V_n_2,V_P_2] :
      ( hBOOL(hAPP(V_P_2,c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_n_2,V_k_2)))
    <=> ( ( V_k_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
         => hBOOL(hAPP(V_P_2,V_n_2)) )
        & ( V_k_2 != c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
         => ! [B_i,B_j] :
              ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_j),V_k_2))
             => ( V_n_2 = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),B_i)),B_j)
               => hBOOL(hAPP(V_P_2,B_j)) ) ) ) ) ) )).

fof(fact_mod__lemma,axiom,(
    ! [V_q,V_b,V_r,V_c] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_c))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_r),V_b))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_b),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_q,V_c))),V_r)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_b),V_c))) ) ) )).

fof(fact_Nat__Transfer_Otransfer__nat__int__functions_I1_J,axiom,(
    ! [V_y,V_x] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_y))
       => hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onat,V_x)),hAPP(c_Int_Onat,V_y)) = hAPP(c_Int_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_x),V_y)) ) ) )).

fof(fact_nat__add__distrib,axiom,(
    ! [V_z_H,V_z] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z_H))
       => hAPP(c_Int_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_z),V_z_H)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onat,V_z)),hAPP(c_Int_Onat,V_z_H)) ) ) )).

fof(fact_power3__eq__cube,axiom,(
    ! [V_a,T_a] :
      ( class_Groups_Omonoid__mult(T_a)
     => hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_a)),V_a) ) )).

fof(fact_int__eq__iff__number__of,axiom,(
    ! [V_v_2,V_m_2] :
      ( hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m_2) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_2)
    <=> ( V_m_2 = hAPP(c_Int_Onat,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_2))
        & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_2))) ) ) )).

fof(fact_Nat__Transfer_Otransfer__nat__int__functions_I2_J,axiom,(
    ! [V_y,V_x] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_y))
       => hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onat,V_x)),hAPP(c_Int_Onat,V_y)) = hAPP(c_Int_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_x),V_y)) ) ) )).

fof(fact_nat__mult__distrib,axiom,(
    ! [V_z_H,V_z] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z))
     => hAPP(c_Int_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_z),V_z_H)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onat,V_z)),hAPP(c_Int_Onat,V_z_H)) ) )).

fof(fact_nat__diff__distrib,axiom,(
    ! [V_z,V_z_H] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z_H))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_z_H),V_z))
       => hAPP(c_Int_Onat,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_z),V_z_H)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Int_Onat,V_z)),hAPP(c_Int_Onat,V_z_H)) ) ) )).

fof(fact_Suc__times__mod__eq,axiom,(
    ! [V_m,V_k] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),V_k))
     => c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_m)),V_k) = c_Groups_Oone__class_Oone(tc_Nat_Onat) ) )).

fof(fact_transfer__nat__int__numerals_I4_J,axiom,(
    hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(c_Int_OBit1(c_Int_OPls))) = hAPP(c_Int_Onat,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit1(c_Int_OBit1(c_Int_OPls)))) )).

fof(fact_power__eq__if,axiom,(
    ! [V_p,V_m] :
      ( ( V_m = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
       => hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_p),V_m) = c_Groups_Oone__class_Oone(tc_Nat_Onat) )
      & ( V_m != c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
       => hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_p),V_m) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_p),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_p),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),c_Groups_Oone__class_Oone(tc_Nat_Onat)))) ) ) )).

fof(fact_zero__less__zpower__abs__iff,axiom,(
    ! [V_n_2,V_x_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_x_2)),V_n_2)))
    <=> ( V_x_2 != c_Groups_Ozero__class_Ozero(tc_Int_Oint)
        | V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) ) )).

fof(fact_Nat__Transfer_Otransfer__nat__int__functions_I3_J,axiom,(
    ! [V_y,V_x] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_y))
       => hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Int_Onat,V_x)),hAPP(c_Int_Onat,V_y)) = hAPP(c_Int_Onat,c_Nat__Transfer_Otsub(V_x,V_y)) ) ) )).

fof(fact_le__div__geq,axiom,(
    ! [V_m,V_n] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n),V_m))
       => hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),V_n) = hAPP(c_Nat_OSuc,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)),V_n)) ) ) )).

fof(fact_split__div__lemma,axiom,(
    ! [V_m_2,V_q_2,V_n_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n_2))
     => ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n_2),V_q_2)),V_m_2))
          & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n_2),hAPP(c_Nat_OSuc,V_q_2)))) )
      <=> V_q_2 = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m_2),V_n_2) ) ) )).

fof(fact_split__div_H,axiom,(
    ! [V_n_2,V_m_2,V_P_2] :
      ( hBOOL(hAPP(V_P_2,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m_2),V_n_2)))
    <=> ( ( V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
          & hBOOL(hAPP(V_P_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) )
        | ? [B_q] :
            ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n_2),B_q)),V_m_2))
            & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n_2),hAPP(c_Nat_OSuc,B_q))))
            & hBOOL(hAPP(V_P_2,B_q)) ) ) ) )).

fof(fact_Suc__div__eq__add3__div__number__of,axiom,(
    ! [V_v,V_m] : hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,V_m)))),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(c_Int_OBit1(c_Int_OPls)))),V_m)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)) )).

fof(fact_Suc__div__eq__add3__div,axiom,(
    ! [V_n,V_m] : hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,V_m)))),V_n) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(c_Int_OBit1(c_Int_OPls)))),V_m)),V_n) )).

fof(fact_div__Suc__eq__div__add3,axiom,(
    ! [V_n,V_m] : hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,V_n)))) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(c_Int_OBit1(c_Int_OPls)))),V_n)) )).

fof(fact_div2__Suc__Suc,axiom,(
    ! [V_m] : hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,V_m))),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(c_Nat_OSuc,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) )).

fof(fact_add__self__div__2,axiom,(
    ! [V_m] : hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_m)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = V_m )).

fof(fact_one__less__nat__eq,axiom,(
    ! [V_z_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),hAPP(c_Int_Onat,V_z_2)))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),V_z_2)) ) )).

fof(fact_nat__less__iff,axiom,(
    ! [V_m_2,V_w_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_w_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onat,V_w_2)),V_m_2))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_w_2),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m_2))) ) ) )).

fof(fact_Suc__nat__eq__nat__zadd1,axiom,(
    ! [V_z] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z))
     => hAPP(c_Nat_OSuc,hAPP(c_Int_Onat,V_z)) = hAPP(c_Int_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),V_z)) ) )).

fof(fact_Suc__mod__eq__add3__mod__number__of,axiom,(
    ! [V_v,V_m] : c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,V_m))),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(c_Int_OBit1(c_Int_OPls)))),V_m),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)) )).

fof(fact_Suc__mod__eq__add3__mod,axiom,(
    ! [V_n,V_m] : c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,V_m))),V_n) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(c_Int_OBit1(c_Int_OPls)))),V_m),V_n) )).

fof(fact_mod__Suc__eq__mod__add3,axiom,(
    ! [V_n,V_m] : c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,V_n)))) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(c_Int_OBit1(c_Int_OPls)))),V_n)) )).

fof(fact_zero__eq__power2,axiom,(
    ! [V_a_2,T_b] :
      ( class_Rings_Oring__1__no__zero__divisors(T_b)
     => ( hAPP(hAPP(c_Power_Opower__class_Opower(T_b),V_a_2),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = c_Groups_Ozero__class_Ozero(T_b)
      <=> V_a_2 = c_Groups_Ozero__class_Ozero(T_b) ) ) )).

fof(fact_zero__power2,axiom,(
    ! [T_a] :
      ( class_Rings_Osemiring__1(T_a)
     => hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = c_Groups_Ozero__class_Ozero(T_a) ) )).

fof(fact_power2__eq__square,axiom,(
    ! [V_a,T_a] :
      ( class_Groups_Omonoid__mult(T_a)
     => hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_a) ) )).

fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,(
    ! [V_x,T_a] :
      ( class_Rings_Ocomm__semiring__1(T_a)
     => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_x) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) ) )).

fof(fact_mod2__Suc__Suc,axiom,(
    ! [V_m] : c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,V_m)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) )).

fof(fact_one__power2,axiom,(
    ! [T_a] :
      ( class_Rings_Osemiring__1(T_a)
     => hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Groups_Oone__class_Oone(T_a)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = c_Groups_Oone__class_Oone(T_a) ) )).

fof(fact_transfer__nat__int__numerals_I3_J,axiom,(
    hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))) = hAPP(c_Int_Onat,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) )).

fof(fact_power2__minus,axiom,(
    ! [V_a,T_a] :
      ( class_Rings_Oring__1(T_a)
     => hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) ) )).

fof(fact_nat__mult__distrib__neg,axiom,(
    ! [V_z_H,V_z] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_z),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
     => hAPP(c_Int_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_z),V_z_H)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onat,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_z))),hAPP(c_Int_Onat,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_z_H))) ) )).

fof(fact_power__even__eq,axiom,(
    ! [V_n,V_a,T_a] :
      ( class_Groups_Omonoid__mult(T_a)
     => hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_n)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) ) )).

fof(fact_power2__abs,axiom,(
    ! [V_a,T_a] :
      ( class_Rings_Olinordered__idom(T_a)
     => hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) ) )).

fof(fact_abs__power2,axiom,(
    ! [V_a,T_a] :
      ( class_Rings_Olinordered__idom(T_a)
     => hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) ) )).

fof(fact_zero__le__power2,axiom,(
    ! [V_a,T_a] :
      ( class_Rings_Olinordered__idom(T_a)
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) ) )).

fof(fact_power2__le__imp__le,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Rings_Olinordered__semidom(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_y),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_y))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y)) ) ) ) )).

fof(fact_power2__eq__imp__eq,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Rings_Olinordered__semidom(T_a)
     => ( hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_y),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_x))
         => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_y))
           => V_x = V_y ) ) ) ) )).

fof(fact_power2__less__0,axiom,(
    ! [V_a,T_a] :
      ( class_Rings_Olinordered__idom(T_a)
     => ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),c_Groups_Ozero__class_Ozero(T_a))) ) )).

fof(fact_zero__less__power2,axiom,(
    ! [V_a_2,T_b] :
      ( class_Rings_Olinordered__idom(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(hAPP(c_Power_Opower__class_Opower(T_b),V_a_2),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))))
      <=> V_a_2 != c_Groups_Ozero__class_Ozero(T_b) ) ) )).

fof(fact_sum__power2__eq__zero__iff,axiom,(
    ! [V_y_2,V_x_2,T_b] :
      ( class_Rings_Olinordered__idom(T_b)
     => ( hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Power_Opower__class_Opower(T_b),V_x_2),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(T_b),V_y_2),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) = c_Groups_Ozero__class_Ozero(T_b)
      <=> ( V_x_2 = c_Groups_Ozero__class_Ozero(T_b)
          & V_y_2 = c_Groups_Ozero__class_Ozero(T_b) ) ) ) )).

fof(fact_nat__2,axiom,(
    hAPP(c_Int_Onat,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) )).

fof(fact_power2__eq__square__number__of,axiom,(
    ! [V_w,T_b] :
      ( ( class_Groups_Omonoid__mult(T_b)
        & class_Int_Onumber(T_b) )
     => hAPP(hAPP(c_Power_Opower__class_Opower(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w)) ) )).

fof(fact_power2__eq__1__iff,axiom,(
    ! [V_a_2,T_b] :
      ( class_Rings_Oring__1__no__zero__divisors(T_b)
     => ( hAPP(hAPP(c_Power_Opower__class_Opower(T_b),V_a_2),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = c_Groups_Oone__class_Oone(T_b)
      <=> ( V_a_2 = c_Groups_Oone__class_Oone(T_b)
          | V_a_2 = hAPP(c_Groups_Ouminus__class_Ouminus(T_b),c_Groups_Oone__class_Oone(T_b)) ) ) ) )).

fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,(
    ! [V_n,V_x,T_a] :
      ( class_Rings_Ocomm__semiring__1(T_a)
     => hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_n)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_n)) ) )).

fof(fact_power__minus__even,axiom,(
    ! [V_n,V_a,T_a] :
      ( class_Rings_Oring__1(T_a)
     => hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_n)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_n)) ) )).

fof(fact_div__2__gt__zero,axiom,(
    ! [V_n] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),V_n))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_n),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) ) )).

fof(fact_power2__less__imp__less,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Rings_Olinordered__semidom(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_y),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_y))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y)) ) ) ) )).

fof(fact_sum__power2__le__zero__iff,axiom,(
    ! [V_y_2,V_x_2,T_b] :
      ( class_Rings_Olinordered__idom(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Power_Opower__class_Opower(T_b),V_x_2),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(T_b),V_y_2),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))),c_Groups_Ozero__class_Ozero(T_b)))
      <=> ( V_x_2 = c_Groups_Ozero__class_Ozero(T_b)
          & V_y_2 = c_Groups_Ozero__class_Ozero(T_b) ) ) ) )).

fof(fact_sum__power2__ge__zero,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Rings_Olinordered__idom(T_a)
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_y),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))))) ) )).

fof(fact_not__sum__power2__lt__zero,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Rings_Olinordered__idom(T_a)
     => ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_y),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))),c_Groups_Ozero__class_Ozero(T_a))) ) )).

fof(fact_sum__power2__gt__zero__iff,axiom,(
    ! [V_y_2,V_x_2,T_b] :
      ( class_Rings_Olinordered__idom(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Power_Opower__class_Opower(T_b),V_x_2),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(T_b),V_y_2),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))))
      <=> ( V_x_2 != c_Groups_Ozero__class_Ozero(T_b)
          | V_y_2 != c_Groups_Ozero__class_Ozero(T_b) ) ) ) )).

fof(fact_power2__sum,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Int_Onumber__ring(T_a)
     => hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_x),V_y)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_y),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_x)),V_y)) ) )).

fof(fact_zero__le__even__power_H,axiom,(
    ! [V_n,V_a,T_a] :
      ( class_Rings_Olinordered__idom(T_a)
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_n)))) ) )).

fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I37_J,axiom,(
    ! [V_n,V_x,T_a] :
      ( class_Rings_Ocomm__semiring__1(T_a)
     => hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_n))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_n)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_n))) ) )).

fof(fact_power__odd__eq,axiom,(
    ! [V_n,V_a,T_a] :
      ( class_Groups_Omonoid__mult(T_a)
     => hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_n))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) ) )).

fof(fact_mod2__gr__0,axiom,(
    ! [V_m_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m_2,hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))))
    <=> c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m_2,hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = c_Groups_Oone__class_Oone(tc_Nat_Onat) ) )).

fof(fact_power__minus1__even,axiom,(
    ! [V_n,T_a] :
      ( class_Rings_Oring__1(T_a)
     => hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),c_Groups_Oone__class_Oone(T_a))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_n)) = c_Groups_Oone__class_Oone(T_a) ) )).

fof(fact_int__power__div__base,axiom,(
    ! [V_k,V_m] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_m))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_k))
       => hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_k),V_m)),V_k) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_k),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) ) ) )).

fof(fact_Suc__n__div__2__gt__zero,axiom,(
    ! [V_n] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),c_Groups_Oone__class_Oone(tc_Nat_Onat))),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) ) )).

fof(fact_power2__diff,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Int_Onumber__ring(T_a)
     => hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_x),V_y)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_y),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_x)),V_y)) ) )).

fof(fact_odd__0__le__power__imp__0__le,axiom,(
    ! [V_n,V_a,T_a] :
      ( class_Rings_Olinordered__idom(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_n)))))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) ) ) )).

fof(fact_int__of__code,axiom,(
    ! [V_k] :
      ( ( V_k = c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)
       => hAPP(c_Code__Numeral_Oint__of,V_k) = c_Groups_Ozero__class_Ozero(tc_Int_Oint) )
      & ( V_k != c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)
       => ( ( c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,V_k,hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)
           => hAPP(c_Code__Numeral_Oint__of,V_k) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(c_Code__Numeral_Oint__of,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),V_k),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) )
          & ( c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,V_k,hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) != c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)
           => hAPP(c_Code__Numeral_Oint__of,V_k) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(c_Code__Numeral_Oint__of,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),V_k),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))))),c_Groups_Oone__class_Oone(tc_Int_Oint)) ) ) ) ) )).

fof(fact_power__strict__mono,axiom,(
    ! [V_n,V_b,V_a,T_a] :
      ( class_Rings_Olinordered__semidom(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a))
         => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n))
           => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_b),V_n))) ) ) ) ) )).

fof(fact_power__diff,axiom,(
    ! [V_m,V_n,V_a,T_a] :
      ( class_Fields_Ofield(T_a)
     => ( V_a != c_Groups_Ozero__class_Ozero(T_a)
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n),V_m))
         => hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_m)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)) ) ) ) )).

fof(fact_one__less__power,axiom,(
    ! [V_n,V_a,T_a] :
      ( class_Rings_Olinordered__semidom(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Oone__class_Oone(T_a)),V_a))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Oone__class_Oone(T_a)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n))) ) ) ) )).

fof(fact_power__increasing__iff,axiom,(
    ! [V_y_2,V_x_2,V_b_2,T_b] :
      ( class_Rings_Olinordered__semidom(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Oone__class_Oone(T_b)),V_b_2))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Power_Opower__class_Opower(T_b),V_b_2),V_x_2)),hAPP(hAPP(c_Power_Opower__class_Opower(T_b),V_b_2),V_y_2)))
        <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_x_2),V_y_2)) ) ) ) )).

fof(fact_power__le__imp__le__exp,axiom,(
    ! [V_n,V_m,V_a,T_a] :
      ( class_Rings_Olinordered__semidom(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Oone__class_Oone(T_a)),V_a))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_m)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),V_n)) ) ) ) )).

fof(fact_field__power__not__zero,axiom,(
    ! [V_n,V_a,T_a] :
      ( class_Rings_Oring__1__no__zero__divisors(T_a)
     => ( V_a != c_Groups_Ozero__class_Ozero(T_a)
       => hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n) != c_Groups_Ozero__class_Ozero(T_a) ) ) )).

fof(fact_nat__power__eq__Suc__0__iff,axiom,(
    ! [V_m_2,V_x_2] :
      ( hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_x_2),V_m_2) = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))
    <=> ( V_m_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
        | V_x_2 = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) ) ) )).

fof(fact_power__Suc__0,axiom,(
    ! [V_n] : hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),V_n) = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) )).

fof(fact_nat__power__less__imp__less,axiom,(
    ! [V_n,V_m,V_i] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_i))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_i),V_m)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_i),V_n)))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) ) ) )).

fof(fact_nat__zero__less__power__iff,axiom,(
    ! [V_n_2,V_x_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_x_2),V_n_2)))
    <=> ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_x_2))
        | V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) ) )).

fof(fact_power__mult__distrib,axiom,(
    ! [V_n,V_b,V_a,T_a] :
      ( class_Groups_Ocomm__monoid__mult(T_a)
     => hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)),V_n) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_b),V_n)) ) )).

fof(fact_power__commutes,axiom,(
    ! [V_n,V_a,T_a] :
      ( class_Groups_Omonoid__mult(T_a)
     => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)),V_a) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)) ) )).

fof(fact_power__one,axiom,(
    ! [V_n,T_a] :
      ( class_Groups_Omonoid__mult(T_a)
     => hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Groups_Oone__class_Oone(T_a)),V_n) = c_Groups_Oone__class_Oone(T_a) ) )).

fof(fact_of__nat__power,axiom,(
    ! [V_n,V_m,T_a] :
      ( class_Rings_Osemiring__1(T_a)
     => hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_m),V_n)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),V_m)),V_n) ) )).

fof(fact_power__mult,axiom,(
    ! [V_n,V_m,V_a,T_a] :
      ( class_Groups_Omonoid__mult(T_a)
     => hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_m)),V_n) ) )).

fof(fact_power__divide,axiom,(
    ! [V_n,V_b,V_a,T_a] :
      ( class_Fields_Ofield__inverse__zero(T_a)
     => hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_b)),V_n) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_b),V_n)) ) )).

fof(fact_power__abs,axiom,(
    ! [V_n,V_a,T_a] :
      ( class_Rings_Olinordered__idom(T_a)
     => hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)),V_n) ) )).

fof(fact_power__one__right,axiom,(
    ! [V_a,T_a] :
      ( class_Groups_Omonoid__mult(T_a)
     => hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = V_a ) )).

fof(fact_zero__le__power,axiom,(
    ! [V_n,V_a,T_a] :
      ( class_Rings_Olinordered__semidom(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n))) ) ) )).

fof(fact_power__mono,axiom,(
    ! [V_n,V_b,V_a,T_a] :
      ( class_Rings_Olinordered__semidom(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_b),V_n))) ) ) ) )).

fof(fact_zero__less__power,axiom,(
    ! [V_n,V_a,T_a] :
      ( class_Rings_Olinordered__semidom(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n))) ) ) )).

fof(fact_power__eq__0__iff,axiom,(
    ! [V_n_2,V_a_2,T_b] :
      ( ( class_Power_Opower(T_b)
        & class_Rings_Omult__zero(T_b)
        & class_Rings_Ono__zero__divisors(T_b)
        & class_Rings_Ozero__neq__one(T_b) )
     => ( hAPP(hAPP(c_Power_Opower__class_Opower(T_b),V_a_2),V_n_2) = c_Groups_Ozero__class_Ozero(T_b)
      <=> ( V_a_2 = c_Groups_Ozero__class_Ozero(T_b)
          & V_n_2 != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) ) ) ) )).

fof(fact_one__le__power,axiom,(
    ! [V_n,V_a,T_a] :
      ( class_Rings_Olinordered__semidom(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Oone__class_Oone(T_a)),V_a))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Oone__class_Oone(T_a)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n))) ) ) )).

fof(fact_power__0__Suc,axiom,(
    ! [V_n,T_a] :
      ( ( class_Power_Opower(T_a)
        & class_Rings_Osemiring__0(T_a) )
     => hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(c_Nat_OSuc,V_n)) = c_Groups_Ozero__class_Ozero(T_a) ) )).

fof(fact_power__inject__exp,axiom,(
    ! [V_n_2,V_m_2,V_a_2,T_b] :
      ( class_Rings_Olinordered__semidom(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Oone__class_Oone(T_b)),V_a_2))
       => ( hAPP(hAPP(c_Power_Opower__class_Opower(T_b),V_a_2),V_m_2) = hAPP(hAPP(c_Power_Opower__class_Opower(T_b),V_a_2),V_n_2)
        <=> V_m_2 = V_n_2 ) ) ) )).

fof(fact_power__Suc2,axiom,(
    ! [V_n,V_a,T_a] :
      ( class_Groups_Omonoid__mult(T_a)
     => hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(c_Nat_OSuc,V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)),V_a) ) )).

fof(fact_power__Suc,axiom,(
    ! [V_n,V_a,T_a] :
      ( class_Power_Opower(T_a)
     => hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(c_Nat_OSuc,V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)) ) )).

fof(fact_nat__one__le__power,axiom,(
    ! [V_n,V_i] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),V_i))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_i),V_n))) ) )).

fof(fact_power__0,axiom,(
    ! [V_a,T_a] :
      ( class_Power_Opower(T_a)
     => hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = c_Groups_Oone__class_Oone(T_a) ) )).

fof(fact_power__add,axiom,(
    ! [V_n,V_m,V_a,T_a] :
      ( class_Groups_Omonoid__mult(T_a)
     => hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_m)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)) ) )).

fof(fact_nonzero__power__divide,axiom,(
    ! [V_n,V_a,V_b,T_a] :
      ( class_Fields_Ofield(T_a)
     => ( V_b != c_Groups_Ozero__class_Ozero(T_a)
       => hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_b)),V_n) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_b),V_n)) ) ) )).

fof(fact_power__one__over,axiom,(
    ! [V_n,V_a,T_a] :
      ( class_Fields_Ofield__inverse__zero(T_a)
     => hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),c_Groups_Oone__class_Oone(T_a)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),c_Groups_Oone__class_Oone(T_a)),V_a)),V_n) ) )).

fof(fact_abs__power__minus,axiom,(
    ! [V_n,V_a,T_a] :
      ( class_Rings_Olinordered__idom(T_a)
     => hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),V_n)) = hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)) ) )).

fof(fact_power__less__imp__less__base,axiom,(
    ! [V_b,V_n,V_a,T_a] :
      ( class_Rings_Olinordered__semidom(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_b),V_n)))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_b))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)) ) ) ) )).

fof(fact_power__le__imp__le__base,axiom,(
    ! [V_b,V_n,V_a,T_a] :
      ( class_Rings_Olinordered__semidom(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(c_Nat_OSuc,V_n))),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_b),hAPP(c_Nat_OSuc,V_n))))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_b))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b)) ) ) ) )).

fof(fact_power__inject__base,axiom,(
    ! [V_b,V_n,V_a,T_a] :
      ( class_Rings_Olinordered__semidom(T_a)
     => ( hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(c_Nat_OSuc,V_n)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_b),hAPP(c_Nat_OSuc,V_n))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a))
         => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_b))
           => V_a = V_b ) ) ) ) )).

fof(fact_power__gt1__lemma,axiom,(
    ! [V_n,V_a,T_a] :
      ( class_Rings_Olinordered__semidom(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Oone__class_Oone(T_a)),V_a))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Oone__class_Oone(T_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)))) ) ) )).

fof(fact_power__less__power__Suc,axiom,(
    ! [V_n,V_a,T_a] :
      ( class_Rings_Olinordered__semidom(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Oone__class_Oone(T_a)),V_a))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)))) ) ) )).

fof(fact_power__0__left,axiom,(
    ! [V_n,T_a] :
      ( ( class_Power_Opower(T_a)
        & class_Rings_Osemiring__0(T_a) )
     => ( ( V_n = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
         => hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_n) = c_Groups_Oone__class_Oone(T_a) )
        & ( V_n != c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
         => hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_n) = c_Groups_Ozero__class_Ozero(T_a) ) ) ) )).

fof(fact_power__gt1,axiom,(
    ! [V_n,V_a,T_a] :
      ( class_Rings_Olinordered__semidom(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Oone__class_Oone(T_a)),V_a))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Oone__class_Oone(T_a)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(c_Nat_OSuc,V_n)))) ) ) )).

fof(fact_power__strict__increasing,axiom,(
    ! [V_a,V_N,V_n,T_a] :
      ( class_Rings_Olinordered__semidom(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),V_N))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Oone__class_Oone(T_a)),V_a))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_N))) ) ) ) )).

fof(fact_power__less__imp__less__exp,axiom,(
    ! [V_n,V_m,V_a,T_a] :
      ( class_Rings_Olinordered__semidom(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Oone__class_Oone(T_a)),V_a))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_m)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) ) ) ) )).

fof(fact_power__strict__increasing__iff,axiom,(
    ! [V_y_2,V_x_2,V_b_2,T_b] :
      ( class_Rings_Olinordered__semidom(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Oone__class_Oone(T_b)),V_b_2))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Power_Opower__class_Opower(T_b),V_b_2),V_x_2)),hAPP(hAPP(c_Power_Opower__class_Opower(T_b),V_b_2),V_y_2)))
        <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_x_2),V_y_2)) ) ) ) )).

fof(fact_zero__le__power__abs,axiom,(
    ! [V_n,V_a,T_a] :
      ( class_Rings_Olinordered__idom(T_a)
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)),V_n))) ) )).

fof(fact_power__increasing,axiom,(
    ! [V_a,V_N,V_n,T_a] :
      ( class_Rings_Olinordered__semidom(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n),V_N))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Oone__class_Oone(T_a)),V_a))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_N))) ) ) ) )).

fof(fact_power__minus,axiom,(
    ! [V_n,V_a,T_a] :
      ( class_Rings_Oring__1(T_a)
     => hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),V_n) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),c_Groups_Oone__class_Oone(T_a))),V_n)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)) ) )).

fof(fact_power__Suc__less,axiom,(
    ! [V_n,V_a,T_a] :
      ( class_Rings_Olinordered__semidom(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),c_Groups_Oone__class_Oone(T_a)))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n))),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n))) ) ) ) )).

fof(fact_power__eq__imp__eq__base,axiom,(
    ! [V_b,V_n,V_a,T_a] :
      ( class_Rings_Olinordered__semidom(T_a)
     => ( hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_b),V_n)
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a))
         => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_b))
           => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n))
             => V_a = V_b ) ) ) ) ) )).

fof(fact_power__Suc__less__one,axiom,(
    ! [V_n,V_a,T_a] :
      ( class_Rings_Olinordered__semidom(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),c_Groups_Oone__class_Oone(T_a)))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(c_Nat_OSuc,V_n))),c_Groups_Oone__class_Oone(T_a))) ) ) ) )).

fof(fact_power__strict__decreasing,axiom,(
    ! [V_a,V_N,V_n,T_a] :
      ( class_Rings_Olinordered__semidom(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),V_N))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a))
         => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),c_Groups_Oone__class_Oone(T_a)))
           => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_N)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n))) ) ) ) ) )).

fof(fact_power__decreasing,axiom,(
    ! [V_a,V_N,V_n,T_a] :
      ( class_Rings_Olinordered__semidom(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n),V_N))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a))
         => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),c_Groups_Oone__class_Oone(T_a)))
           => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_N)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n))) ) ) ) ) )).

fof(fact_nat__aux__def,axiom,(
    ! [V_n,V_i] : c_Int_Onat__aux(V_i,V_n) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onat,V_i)),V_n) )).

fof(fact_transfer__morphism__nat__int,axiom,(
    c_Nat__Transfer_Otransfer__morphism(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat,hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) )).

fof(fact_power__power__power,axiom,(
    ! [T_b] :
      ( class_Power_Opower(T_b)
     => c_Power_Opower__class_Opower(T_b) = c_Power_Opower_Opower(T_b,c_Groups_Oone__class_Oone(T_b),c_Groups_Otimes__class_Otimes(T_b)) ) )).

fof(fact_power__m1__odd,axiom,(
    ! [V_n,T_a] :
      ( class_Int_Onumber__ring(T_a)
     => hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OMin)),hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_n))) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OMin) ) )).

fof(fact_power__m1__even,axiom,(
    ! [V_n,T_a] :
      ( class_Int_Onumber__ring(T_a)
     => hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OMin)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_n)) = c_Groups_Oone__class_Oone(T_a) ) )).

fof(fact_rel__simps_I24_J,axiom,(
    hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OMin),c_Int_OMin)) )).

fof(fact_rel__simps_I7_J,axiom,(
    ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OMin),c_Int_OMin)) )).

fof(fact_rel__simps_I37_J,axiom,(
    c_Int_OPls != c_Int_OMin )).

fof(fact_rel__simps_I40_J,axiom,(
    c_Int_OMin != c_Int_OPls )).

fof(fact_rel__simps_I47_J,axiom,(
    ! [V_k_2] :
      ( c_Int_OBit1(V_k_2) = c_Int_OMin
    <=> V_k_2 = c_Int_OMin ) )).

fof(fact_rel__simps_I43_J,axiom,(
    ! [V_l_2] :
      ( c_Int_OMin = c_Int_OBit1(V_l_2)
    <=> c_Int_OMin = V_l_2 ) )).

fof(fact_Bit1__Min,axiom,(
    c_Int_OBit1(c_Int_OMin) = c_Int_OMin )).

fof(fact_rel__simps_I45_J,axiom,(
    ! [V_k] : c_Int_OBit0(V_k) != c_Int_OMin )).

fof(fact_rel__simps_I42_J,axiom,(
    ! [V_l] : c_Int_OMin != c_Int_OBit0(V_l) )).

fof(fact_bin__less__0__simps_I2_J,axiom,(
    hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OMin),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) )).

fof(fact_rel__simps_I20_J,axiom,(
    ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OPls),c_Int_OMin)) )).

fof(fact_rel__simps_I23_J,axiom,(
    hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OMin),c_Int_OPls)) )).

fof(fact_rel__simps_I30_J,axiom,(
    ! [V_k_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit1(V_k_2)),c_Int_OMin))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k_2),c_Int_OMin)) ) )).

fof(fact_rel__simps_I26_J,axiom,(
    ! [V_k_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OMin),c_Int_OBit1(V_k_2)))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OMin),V_k_2)) ) )).

fof(fact_rel__simps_I3_J,axiom,(
    ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OPls),c_Int_OMin)) )).

fof(fact_rel__simps_I6_J,axiom,(
    hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OMin),c_Int_OPls)) )).

fof(fact_rel__simps_I13_J,axiom,(
    ! [V_k_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit1(V_k_2)),c_Int_OMin))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),c_Int_OMin)) ) )).

fof(fact_rel__simps_I9_J,axiom,(
    ! [V_k_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OMin),c_Int_OBit1(V_k_2)))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OMin),V_k_2)) ) )).

fof(fact_rel__simps_I28_J,axiom,(
    ! [V_k_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit0(V_k_2)),c_Int_OMin))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k_2),c_Int_OMin)) ) )).

fof(fact_rel__simps_I8_J,axiom,(
    ! [V_k_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OMin),c_Int_OBit0(V_k_2)))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OMin),V_k_2)) ) )).

fof(fact_eq__number__of__Pls__Min,axiom,(
    hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OPls) != hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin) )).

fof(fact_Int_OMin__def,axiom,(
    c_Int_OMin = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)) )).

fof(fact_mult__Min,axiom,(
    ! [V_k] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Int_OMin),V_k) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_k) )).

fof(fact_neg__number__of__Min,axiom,(
    hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin))) )).

fof(fact_pred__Pls,axiom,(
    c_Int_Opred(c_Int_OPls) = c_Int_OMin )).

fof(fact_add__Min,axiom,(
    ! [V_k] : hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Int_OMin),V_k) = c_Int_Opred(V_k) )).

fof(fact_add__Min__right,axiom,(
    ! [V_k] : hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_k),c_Int_OMin) = c_Int_Opred(V_k) )).

fof(fact_pred__Min,axiom,(
    c_Int_Opred(c_Int_OMin) = c_Int_OBit0(c_Int_OMin) )).

fof(fact_nonzero__number__of__Min,axiom,(
    ! [T_a] :
      ( class_Int_Onumber__ring(T_a)
     => ~ c_Int_Oiszero(T_a,hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OMin)) ) )).

fof(fact_succ__Min,axiom,(
    c_Int_Osucc(c_Int_OMin) = c_Int_OPls )).

fof(fact_diff__bin__simps_I2_J,axiom,(
    ! [V_k] : hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_k),c_Int_OMin) = c_Int_Osucc(V_k) )).

fof(fact_mult__minus1,axiom,(
    ! [V_z,T_a] :
      ( class_Int_Onumber__ring(T_a)
     => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OMin)),V_z) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_z) ) )).

fof(fact_mult__minus1__right,axiom,(
    ! [V_z,T_a] :
      ( class_Int_Onumber__ring(T_a)
     => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_z),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OMin)) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_z) ) )).

fof(fact_number__of__Min,axiom,(
    ! [T_a] :
      ( class_Int_Onumber__ring(T_a)
     => hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OMin) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),c_Groups_Oone__class_Oone(T_a)) ) )).

fof(fact_arith__simps_I31_J,axiom,(
    ! [T_a] :
      ( class_Int_Onumber__ring(T_a)
     => hAPP(c_Groups_Ouminus__class_Ouminus(T_a),c_Groups_Oone__class_Oone(T_a)) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OMin) ) )).

fof(fact_abs__minus__one,axiom,(
    ! [T_a] :
      ( ( class_Int_Onumber__ring(T_a)
        & class_Rings_Olinordered__idom(T_a) )
     => hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OMin)) = c_Groups_Oone__class_Oone(T_a) ) )).

fof(fact_divide__minus1,axiom,(
    ! [V_x,T_a] :
      ( ( class_Fields_Ofield__inverse__zero(T_a)
        & class_Int_Onumber__ring(T_a) )
     => hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OMin)) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_x) ) )).

fof(fact_rel__simps_I11_J,axiom,(
    ! [V_k_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit0(V_k_2)),c_Int_OMin))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k_2),c_Int_OMin)) ) )).

fof(fact_rel__simps_I25_J,axiom,(
    ! [V_k_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OMin),c_Int_OBit0(V_k_2)))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OMin),V_k_2)) ) )).

fof(fact_zmod__minus1__right,axiom,(
    ! [V_a] : c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin)) = c_Groups_Ozero__class_Ozero(tc_Int_Oint) )).

fof(fact_diff__bin__simps_I4_J,axiom,(
    ! [V_l] : hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Int_OPls),c_Int_OBit1(V_l)) = c_Int_OBit1(hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Int_OMin),V_l)) )).

fof(fact_minus__Min,axiom,(
    hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Int_OMin) = c_Int_OBit1(c_Int_OPls) )).

fof(fact_pos__zmult__eq__1__iff__lemma,axiom,(
    ! [V_n,V_m] :
      ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_m),V_n) = c_Groups_Oone__class_Oone(tc_Int_Oint)
     => ( V_m = c_Groups_Oone__class_Oone(tc_Int_Oint)
        | V_m = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin) ) ) )).

fof(fact_zmult__eq__1__iff,axiom,(
    ! [V_n_2,V_m_2] :
      ( hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_m_2),V_n_2) = c_Groups_Oone__class_Oone(tc_Int_Oint)
    <=> ( ( V_m_2 = c_Groups_Oone__class_Oone(tc_Int_Oint)
          & V_n_2 = c_Groups_Oone__class_Oone(tc_Int_Oint) )
        | ( V_m_2 = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin)
          & V_n_2 = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin) ) ) ) )).

fof(fact_diff__bin__simps_I6_J,axiom,(
    ! [V_l] : hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Int_OMin),c_Int_OBit1(V_l)) = c_Int_OBit0(hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Int_OMin),V_l)) )).

fof(fact_diff__bin__simps_I5_J,axiom,(
    ! [V_l] : hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Int_OMin),c_Int_OBit0(V_l)) = c_Int_OBit1(hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Int_OMin),V_l)) )).

fof(fact_power_Opower_Opower__0,axiom,(
    ! [V_a_2,V_times_2,V_one_2,T_b] : hAPP(hAPP(c_Power_Opower_Opower(T_b,V_one_2,V_times_2),V_a_2),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = V_one_2 )).

fof(fact_power_Opower_Opower__Suc,axiom,(
    ! [V_n_2,V_a_2,V_times_2,V_one_2,T_b] : hAPP(hAPP(c_Power_Opower_Opower(T_b,V_one_2,V_times_2),V_a_2),hAPP(c_Nat_OSuc,V_n_2)) = hAPP(hAPP(V_times_2,V_a_2),hAPP(hAPP(c_Power_Opower_Opower(T_b,V_one_2,V_times_2),V_a_2),V_n_2)) )).

fof(fact_zdiv__minus1__right,axiom,(
    ! [V_a] : hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin)) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a) )).

fof(fact_minus1__divide,axiom,(
    ! [V_x,T_a] :
      ( ( class_Fields_Ofield__inverse__zero(T_a)
        & class_Int_Onumber__ring(T_a) )
     => hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OMin)),V_x) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),c_Groups_Oone__class_Oone(T_a)),V_x)) ) )).

fof(fact_abs__power__minus__one,axiom,(
    ! [V_n,T_a] :
      ( ( class_Int_Onumber__ring(T_a)
        & class_Rings_Olinordered__idom(T_a) )
     => hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OMin)),V_n)) = c_Groups_Oone__class_Oone(T_a) ) )).

fof(fact_div__eq__minus1,axiom,(
    ! [V_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b))
     => hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin)),V_b) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin) ) )).

fof(fact_div__pos__neg__trivial,axiom,(
    ! [V_b,V_a] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a),V_b)),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
       => hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin) ) ) )).

fof(fact_zmod__minus1,axiom,(
    ! [V_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b))
     => c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin),V_b) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_b),c_Groups_Oone__class_Oone(tc_Int_Oint)) ) )).

fof(fact_code__numeral_Osize_I1_J,axiom,(
    c_Code__Numeral_Ocode__numeral_Ocode__numeral__size(c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )).

fof(fact_of__int__of__nat,axiom,(
    ! [V_k,T_a] :
      ( class_Rings_Oring__1(T_a)
     => ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
         => hAPP(c_Int_Oring__1__class_Oof__int(T_a),V_k) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),hAPP(c_Int_Onat,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_k)))) )
        & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
         => hAPP(c_Int_Oring__1__class_Oof__int(T_a),V_k) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),hAPP(c_Int_Onat,V_k)) ) ) ) )).

fof(fact_Nitpick_Oint__lcm__def,axiom,(
    ! [V_y,V_x] : c_Nitpick_Oint__lcm(V_x,V_y) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),c_Nitpick_Onat__lcm(hAPP(c_Int_Onat,hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_x)),hAPP(c_Int_Onat,hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_y)))) )).

fof(fact_negD,axiom,(
    ! [V_x] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_x),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
     => ? [B_n] : V_x = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Nat_OSuc,B_n))) ) )).

fof(fact_code__numeral_Osize_I3_J,axiom,(
    hAPP(c_Nat_Osize__class_Osize(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )).

fof(fact_of__int__eq__iff,axiom,(
    ! [V_z_2,V_w_2,T_b] :
      ( class_Int_Oring__char__0(T_b)
     => ( hAPP(c_Int_Oring__1__class_Oof__int(T_b),V_w_2) = hAPP(c_Int_Oring__1__class_Oof__int(T_b),V_z_2)
      <=> V_w_2 = V_z_2 ) ) )).

fof(fact_number__of__eq,axiom,(
    ! [V_k,T_a] :
      ( class_Int_Onumber__ring(T_a)
     => hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_k) = hAPP(c_Int_Oring__1__class_Oof__int(T_a),V_k) ) )).

fof(fact_int__number__of__def,axiom,(
    ! [V_w] : hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w) = hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),V_w) )).

fof(fact_of__int__int__eq,axiom,(
    ! [V_n] : hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n) )).

fof(fact_of__int__eq__0__iff,axiom,(
    ! [V_z_2,T_b] :
      ( class_Int_Oring__char__0(T_b)
     => ( hAPP(c_Int_Oring__1__class_Oof__int(T_b),V_z_2) = c_Groups_Ozero__class_Ozero(T_b)
      <=> V_z_2 = c_Groups_Ozero__class_Ozero(tc_Int_Oint) ) ) )).

fof(fact_of__int__0__eq__iff,axiom,(
    ! [V_z_2,T_b] :
      ( class_Int_Oring__char__0(T_b)
     => ( c_Groups_Ozero__class_Ozero(T_b) = hAPP(c_Int_Oring__1__class_Oof__int(T_b),V_z_2)
      <=> V_z_2 = c_Groups_Ozero__class_Ozero(tc_Int_Oint) ) ) )).

fof(fact_of__int__0,axiom,(
    ! [T_a] :
      ( class_Rings_Oring__1(T_a)
     => hAPP(c_Int_Oring__1__class_Oof__int(T_a),c_Groups_Ozero__class_Ozero(tc_Int_Oint)) = c_Groups_Ozero__class_Ozero(T_a) ) )).

fof(fact_of__int__le__iff,axiom,(
    ! [V_z_2,V_w_2,T_b] :
      ( class_Rings_Olinordered__idom(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Int_Oring__1__class_Oof__int(T_b),V_w_2)),hAPP(c_Int_Oring__1__class_Oof__int(T_b),V_z_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_w_2),V_z_2)) ) ) )).

fof(fact_of__int__less__iff,axiom,(
    ! [V_z_2,V_w_2,T_b] :
      ( class_Rings_Olinordered__idom(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(c_Int_Oring__1__class_Oof__int(T_b),V_w_2)),hAPP(c_Int_Oring__1__class_Oof__int(T_b),V_z_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_w_2),V_z_2)) ) ) )).

fof(fact_of__int__add,axiom,(
    ! [V_z,V_w,T_a] :
      ( class_Rings_Oring__1(T_a)
     => hAPP(c_Int_Oring__1__class_Oof__int(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_w),V_z)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Int_Oring__1__class_Oof__int(T_a),V_w)),hAPP(c_Int_Oring__1__class_Oof__int(T_a),V_z)) ) )).

fof(fact_of__int__number__of__eq,axiom,(
    ! [V_v,T_a] :
      ( class_Int_Onumber__ring(T_a)
     => hAPP(c_Int_Oring__1__class_Oof__int(T_a),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_v) ) )).

fof(fact_of__int__mult,axiom,(
    ! [V_z,V_w,T_a] :
      ( class_Rings_Oring__1(T_a)
     => hAPP(c_Int_Oring__1__class_Oof__int(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_w),V_z)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Int_Oring__1__class_Oof__int(T_a),V_w)),hAPP(c_Int_Oring__1__class_Oof__int(T_a),V_z)) ) )).

fof(fact_of__int__1,axiom,(
    ! [T_a] :
      ( class_Rings_Oring__1(T_a)
     => hAPP(c_Int_Oring__1__class_Oof__int(T_a),c_Groups_Oone__class_Oone(tc_Int_Oint)) = c_Groups_Oone__class_Oone(T_a) ) )).

fof(fact_of__int__diff,axiom,(
    ! [V_z,V_w,T_a] :
      ( class_Rings_Oring__1(T_a)
     => hAPP(c_Int_Oring__1__class_Oof__int(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_w),V_z)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(c_Int_Oring__1__class_Oof__int(T_a),V_w)),hAPP(c_Int_Oring__1__class_Oof__int(T_a),V_z)) ) )).

fof(fact_of__int__of__nat__eq,axiom,(
    ! [V_n,T_a] :
      ( class_Rings_Oring__1(T_a)
     => hAPP(c_Int_Oring__1__class_Oof__int(T_a),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),V_n) ) )).

fof(fact_of__int__minus,axiom,(
    ! [V_z,T_a] :
      ( class_Rings_Oring__1(T_a)
     => hAPP(c_Int_Oring__1__class_Oof__int(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_z)) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(c_Int_Oring__1__class_Oof__int(T_a),V_z)) ) )).

fof(fact_Ints__of__int,axiom,(
    ! [V_z_2,T_b] :
      ( class_Rings_Oring__1(T_b)
     => hBOOL(hAPP(hAPP(c_member(T_b),hAPP(c_Int_Oring__1__class_Oof__int(T_b),V_z_2)),c_Int_Oring__1__class_OInts(T_b))) ) )).

fof(fact_of__int__power,axiom,(
    ! [V_n,V_z,T_a] :
      ( class_Rings_Oring__1(T_a)
     => hAPP(c_Int_Oring__1__class_Oof__int(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_z),V_n)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(c_Int_Oring__1__class_Oof__int(T_a),V_z)),V_n) ) )).

fof(fact_of__int__m1,axiom,(
    ! [T_a] :
      ( class_Int_Onumber__ring(T_a)
     => hAPP(c_Int_Oring__1__class_Oof__int(T_a),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin)) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OMin) ) )).

fof(fact_of__int__0__le__iff,axiom,(
    ! [V_z_2,T_b] :
      ( class_Rings_Olinordered__idom(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(c_Int_Oring__1__class_Oof__int(T_b),V_z_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z_2)) ) ) )).

fof(fact_of__int__le__0__iff,axiom,(
    ! [V_z_2,T_b] :
      ( class_Rings_Olinordered__idom(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Int_Oring__1__class_Oof__int(T_b),V_z_2)),c_Groups_Ozero__class_Ozero(T_b)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_z_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) ) ) )).

fof(fact_of__int__0__less__iff,axiom,(
    ! [V_z_2,T_b] :
      ( class_Rings_Olinordered__idom(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(c_Int_Oring__1__class_Oof__int(T_b),V_z_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z_2)) ) ) )).

fof(fact_of__int__less__0__iff,axiom,(
    ! [V_z_2,T_b] :
      ( class_Rings_Olinordered__idom(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(c_Int_Oring__1__class_Oof__int(T_b),V_z_2)),c_Groups_Ozero__class_Ozero(T_b)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_z_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) ) ) )).

fof(fact_of__nat__nat,axiom,(
    ! [V_z,T_a] :
      ( class_Rings_Oring__1(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z))
       => hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),hAPP(c_Int_Onat,V_z)) = hAPP(c_Int_Oring__1__class_Oof__int(T_a),V_z) ) ) )).

fof(fact_Nitpick_Onat__lcm__def,axiom,(
    ! [V_y,V_x] : c_Nitpick_Onat__lcm(V_x,V_y) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_x),V_y)),c_Nitpick_Onat__gcd(V_x,V_y)) )).

fof(fact_code__numeral_Osize_I4_J,axiom,(
    ! [V_code__numeral] : hAPP(c_Nat_Osize__class_Osize(tc_Code__Numeral_Ocode__numeral),c_Code__Numeral_OSuc__code__numeral(V_code__numeral)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_Code__Numeral_Ocode__numeral),V_code__numeral)),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) )).

fof(fact_code__numeral_Osize_I2_J,axiom,(
    ! [V_code__numeral] : c_Code__Numeral_Ocode__numeral_Ocode__numeral__size(c_Code__Numeral_OSuc__code__numeral(V_code__numeral)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Code__Numeral_Ocode__numeral_Ocode__numeral__size(V_code__numeral)),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) )).

fof(fact_card__Pow,axiom,(
    ! [V_A_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
     => hAPP(c_Finite__Set_Ocard(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Set_OPow(T_b),V_A_2)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))),hAPP(c_Finite__Set_Ocard(T_b),V_A_2)) ) )).

fof(fact_sgn__if,axiom,(
    ! [V_x,T_a] :
      ( class_Groups_Osgn__if(T_a)
     => ( ( V_x = c_Groups_Ozero__class_Ozero(T_a)
         => c_Groups_Osgn__class_Osgn(T_a,V_x) = c_Groups_Ozero__class_Ozero(T_a) )
        & ( V_x != c_Groups_Ozero__class_Ozero(T_a)
         => ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_x))
             => c_Groups_Osgn__class_Osgn(T_a,V_x) = c_Groups_Oone__class_Oone(T_a) )
            & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_x))
             => c_Groups_Osgn__class_Osgn(T_a,V_x) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),c_Groups_Oone__class_Oone(T_a)) ) ) ) ) ) )).

fof(fact_PowI,axiom,(
    ! [V_B_2,V_A_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))
     => hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(c_Set_OPow(T_b),V_B_2))) ) )).

fof(fact_sgn__sgn,axiom,(
    ! [V_a,T_a] :
      ( class_Rings_Olinordered__idom(T_a)
     => c_Groups_Osgn__class_Osgn(T_a,c_Groups_Osgn__class_Osgn(T_a,V_a)) = c_Groups_Osgn__class_Osgn(T_a,V_a) ) )).

fof(fact_Pow__top,axiom,(
    ! [V_A_2,T_b] : hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(c_Set_OPow(T_b),V_A_2))) )).

fof(fact_Pow__not__empty,axiom,(
    ! [V_A_2,T_b] : hAPP(c_Set_OPow(T_b),V_A_2) != c_Orderings_Obot__class_Obot(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)) )).

fof(fact_code__numeral_Oinject,axiom,(
    ! [V_code__numeral_H_2,V_code__numeral_2] :
      ( c_Code__Numeral_OSuc__code__numeral(V_code__numeral_2) = c_Code__Numeral_OSuc__code__numeral(V_code__numeral_H_2)
    <=> V_code__numeral_2 = V_code__numeral_H_2 ) )).

fof(fact_Pow__empty,axiom,(
    ! [T_b] : hAPP(c_Set_OPow(T_b),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) = hAPP(hAPP(c_Set_Oinsert(tc_fun(T_b,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))),c_Orderings_Obot__class_Obot(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool))) )).

fof(fact_Pow__bottom,axiom,(
    ! [V_B_2,T_b] : hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))),hAPP(c_Set_OPow(T_b),V_B_2))) )).

fof(fact_finite__Pow__iff,axiom,(
    ! [V_A_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Set_OPow(T_b),V_A_2)))
    <=> hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) ) )).

fof(fact_PowD,axiom,(
    ! [V_B_2,V_A_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(c_Set_OPow(T_b),V_B_2)))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) ) )).

fof(fact_Pow__mono,axiom,(
    ! [V_B_2,V_A_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)),hAPP(c_Set_OPow(T_b),V_A_2)),hAPP(c_Set_OPow(T_b),V_B_2))) ) )).

fof(fact_Pow__iff,axiom,(
    ! [V_B_2,V_A_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(c_Set_OPow(T_b),V_B_2)))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) ) )).

fof(fact_sgn__0__0,axiom,(
    ! [V_a_2,T_b] :
      ( class_Rings_Olinordered__idom(T_b)
     => ( c_Groups_Osgn__class_Osgn(T_b,V_a_2) = c_Groups_Ozero__class_Ozero(T_b)
      <=> V_a_2 = c_Groups_Ozero__class_Ozero(T_b) ) ) )).

fof(fact_sgn0,axiom,(
    ! [T_a] :
      ( class_Groups_Osgn__if(T_a)
     => c_Groups_Osgn__class_Osgn(T_a,c_Groups_Ozero__class_Ozero(T_a)) = c_Groups_Ozero__class_Ozero(T_a) ) )).

fof(fact_sgn__times,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Rings_Olinordered__idom(T_a)
     => c_Groups_Osgn__class_Osgn(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Osgn__class_Osgn(T_a,V_a)),c_Groups_Osgn__class_Osgn(T_a,V_b)) ) )).

fof(fact_code__numeral_Osimps_I2_J,axiom,(
    ! [V_code__numeral_H] : c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != c_Code__Numeral_OSuc__code__numeral(V_code__numeral_H) )).

fof(fact_code__numeral_Osimps_I3_J,axiom,(
    ! [V_code__numeral_H_1] : c_Code__Numeral_OSuc__code__numeral(V_code__numeral_H_1) != c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) )).

fof(fact_sgn__less,axiom,(
    ! [V_a_2,T_b] :
      ( class_Rings_Olinordered__idom(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Osgn__class_Osgn(T_b,V_a_2)),c_Groups_Ozero__class_Ozero(T_b)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),c_Groups_Ozero__class_Ozero(T_b))) ) ) )).

fof(fact_sgn__greater,axiom,(
    ! [V_a_2,T_b] :
      ( class_Rings_Olinordered__idom(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),c_Groups_Osgn__class_Osgn(T_b,V_a_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_a_2)) ) ) )).

fof(fact_mult__sgn__abs,axiom,(
    ! [V_x,T_a] :
      ( class_Rings_Olinordered__idom(T_a)
     => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Osgn__class_Osgn(T_a,V_x)),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_x)) = V_x ) )).

fof(fact_abs__sgn,axiom,(
    ! [V_k,T_a] :
      ( class_Rings_Olinordered__idom(T_a)
     => hAPP(c_Groups_Oabs__class_Oabs(T_a),V_k) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_k),c_Groups_Osgn__class_Osgn(T_a,V_k)) ) )).

fof(fact_Suc__code__numeral__minus__one,axiom,(
    ! [V_n] : hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Code__Numeral_Ocode__numeral),c_Code__Numeral_OSuc__code__numeral(V_n)),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)) = V_n )).

fof(fact_sgn__pos,axiom,(
    ! [V_a,T_a] :
      ( class_Rings_Olinordered__idom(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a))
       => c_Groups_Osgn__class_Osgn(T_a,V_a) = c_Groups_Oone__class_Oone(T_a) ) ) )).

fof(fact_sgn__1__pos,axiom,(
    ! [V_a_2,T_b] :
      ( class_Rings_Olinordered__idom(T_b)
     => ( c_Groups_Osgn__class_Osgn(T_b,V_a_2) = c_Groups_Oone__class_Oone(T_b)
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_a_2)) ) ) )).

fof(fact_nat__gcd_Osimps,axiom,(
    ! [V_x,V_y] :
      ( ( V_y = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
       => c_Nitpick_Onat__gcd(V_x,V_y) = V_x )
      & ( V_y != c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
       => c_Nitpick_Onat__gcd(V_x,V_y) = c_Nitpick_Onat__gcd(V_y,c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_x,V_y)) ) ) )).

fof(fact_zsgn__def,axiom,(
    ! [V_i] :
      ( ( V_i = c_Groups_Ozero__class_Ozero(tc_Int_Oint)
       => c_Groups_Osgn__class_Osgn(tc_Int_Oint,V_i) = c_Groups_Ozero__class_Ozero(tc_Int_Oint) )
      & ( V_i != c_Groups_Ozero__class_Ozero(tc_Int_Oint)
       => ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_i))
           => c_Groups_Osgn__class_Osgn(tc_Int_Oint,V_i) = c_Groups_Oone__class_Oone(tc_Int_Oint) )
          & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_i))
           => c_Groups_Osgn__class_Osgn(tc_Int_Oint,V_i) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)) ) ) ) ) )).

fof(fact_sgn__neg,axiom,(
    ! [V_a,T_a] :
      ( class_Rings_Olinordered__idom(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a)))
       => c_Groups_Osgn__class_Osgn(T_a,V_a) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),c_Groups_Oone__class_Oone(T_a)) ) ) )).

fof(fact_sgn__1__neg,axiom,(
    ! [V_a_2,T_b] :
      ( class_Rings_Olinordered__idom(T_b)
     => ( c_Groups_Osgn__class_Osgn(T_b,V_a_2) = hAPP(c_Groups_Ouminus__class_Ouminus(T_b),c_Groups_Oone__class_Oone(T_b))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),c_Groups_Ozero__class_Ozero(T_b))) ) ) )).

fof(fact_Nitpick_Oint__gcd__def,axiom,(
    ! [V_y,V_x] : hAPP(hAPP(c_Nitpick_Oint__gcd,V_x),V_y) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),c_Nitpick_Onat__gcd(hAPP(c_Int_Onat,hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_x)),hAPP(c_Int_Onat,hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_y)))) )).

fof(fact_code__numeral_Osimps_I5_J,axiom,(
    ! [V_code__numeral_2,V_f2_2,V_f1_2,T_b] : c_Code__Numeral_Ocode__numeral_Ocode__numeral__case(T_b,V_f1_2,V_f2_2,c_Code__Numeral_OSuc__code__numeral(V_code__numeral_2)) = hAPP(V_f2_2,V_code__numeral_2) )).

fof(fact_nat0__intermed__int__val,axiom,(
    ! [V_k_2,V_f_2,V_n_2] :
      ( ! [B_i] :
          ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_i),V_n_2))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(V_f_2,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B_i),c_Groups_Oone__class_Oone(tc_Nat_Onat)))),hAPP(V_f_2,B_i)))),c_Groups_Oone__class_Oone(tc_Int_Oint))) )
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(V_f_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),V_k_2))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k_2),hAPP(V_f_2,V_n_2)))
         => ? [B_i] :
              ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B_i),V_n_2))
              & hAPP(V_f_2,B_i) = V_k_2 ) ) ) ) )).

fof(fact_Nat__Transfer_Otransfer__int__nat__function__closures_I7_J,axiom,(
    c_Nat__Transfer_Ois__nat(hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) )).

fof(fact_code__numeral_Osimps_I4_J,axiom,(
    ! [V_f2_2,V_f1_2,T_b] : c_Code__Numeral_Ocode__numeral_Ocode__numeral__case(T_b,V_f1_2,V_f2_2,c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)) = V_f1_2 )).

fof(fact_Nat__Transfer_Otransfer__int__nat__function__closures_I5_J,axiom,(
    c_Nat__Transfer_Ois__nat(c_Groups_Ozero__class_Ozero(tc_Int_Oint)) )).

fof(fact_Nat__Transfer_Otransfer__int__nat__function__closures_I1_J,axiom,(
    ! [V_y,V_x] :
      ( c_Nat__Transfer_Ois__nat(V_x)
     => ( c_Nat__Transfer_Ois__nat(V_y)
       => c_Nat__Transfer_Ois__nat(hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_x),V_y)) ) ) )).

fof(fact_Nat__Transfer_Otransfer__int__nat__function__closures_I9_J,axiom,(
    ! [V_z] : c_Nat__Transfer_Ois__nat(hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_z)) )).

fof(fact_Nat__Transfer_Otransfer__int__nat__function__closures_I6_J,axiom,(
    c_Nat__Transfer_Ois__nat(c_Groups_Oone__class_Oone(tc_Int_Oint)) )).

fof(fact_Nat__Transfer_Otransfer__int__nat__function__closures_I2_J,axiom,(
    ! [V_y,V_x] :
      ( c_Nat__Transfer_Ois__nat(V_x)
     => ( c_Nat__Transfer_Ois__nat(V_y)
       => c_Nat__Transfer_Ois__nat(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_x),V_y)) ) ) )).

fof(fact_Divides_Otransfer__int__nat__function__closures_I2_J,axiom,(
    ! [V_y,V_x] :
      ( c_Nat__Transfer_Ois__nat(V_x)
     => ( c_Nat__Transfer_Ois__nat(V_y)
       => c_Nat__Transfer_Ois__nat(c_Divides_Odiv__class_Omod(tc_Int_Oint,V_x,V_y)) ) ) )).

fof(fact_Divides_Otransfer__int__nat__function__closures_I1_J,axiom,(
    ! [V_y,V_x] :
      ( c_Nat__Transfer_Ois__nat(V_x)
     => ( c_Nat__Transfer_Ois__nat(V_y)
       => c_Nat__Transfer_Ois__nat(hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_x),V_y)) ) ) )).

fof(fact_Nat__Transfer_Otransfer__int__nat__function__closures_I4_J,axiom,(
    ! [V_n,V_x] :
      ( c_Nat__Transfer_Ois__nat(V_x)
     => c_Nat__Transfer_Ois__nat(hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_x),V_n)) ) )).

fof(fact_Nat__Transfer_Otransfer__int__nat__function__closures_I3_J,axiom,(
    ! [V_y,V_x] :
      ( c_Nat__Transfer_Ois__nat(V_x)
     => ( c_Nat__Transfer_Ois__nat(V_y)
       => c_Nat__Transfer_Ois__nat(c_Nat__Transfer_Otsub(V_x,V_y)) ) ) )).

fof(fact_is__nat__def,axiom,(
    ! [V_x_2] :
      ( c_Nat__Transfer_Ois__nat(V_x_2)
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x_2)) ) )).

fof(fact_Nat__Transfer_Otransfer__int__nat__function__closures_I8_J,axiom,(
    c_Nat__Transfer_Ois__nat(hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit1(c_Int_OBit1(c_Int_OPls)))) )).

fof(fact_Powp__mono,axiom,(
    ! [V_B_2,V_A_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)),c_Predicate_OPowp(T_b,V_A_2)),c_Predicate_OPowp(T_b,V_B_2))) ) )).

fof(fact_nat__mod__eq__lemma,axiom,(
    ! [V_y,V_n,V_x] :
      ( c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_x,V_n) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_y,V_n)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_y),V_x))
       => ? [B_q] : V_x = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_y),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n),B_q)) ) ) )).

fof(fact_mod__induct__0,axiom,(
    ! [V_i_2,V_P_2,V_p_2] :
      ( ! [B_i] :
          ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_i),V_p_2))
         => ( hBOOL(hAPP(V_P_2,B_i))
           => hBOOL(hAPP(V_P_2,c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Nat_OSuc,B_i),V_p_2))) ) )
     => ( hBOOL(hAPP(V_P_2,V_i_2))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i_2),V_p_2))
         => hBOOL(hAPP(V_P_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) ) ) ) )).

fof(fact_nat__intermed__int__val,axiom,(
    ! [V_k_2,V_f_2,V_n_2,V_m_2] :
      ( ! [B_i] :
          ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),B_i))
            & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_i),V_n_2)) )
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(V_f_2,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B_i),c_Groups_Oone__class_Oone(tc_Nat_Onat)))),hAPP(V_f_2,B_i)))),c_Groups_Oone__class_Oone(tc_Int_Oint))) )
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(V_f_2,V_m_2)),V_k_2))
         => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k_2),hAPP(V_f_2,V_n_2)))
           => ? [B_i] :
                ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),B_i))
                & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B_i),V_n_2))
                & hAPP(V_f_2,B_i) = V_k_2 ) ) ) ) ) )).

fof(fact_log_Osimps,axiom,(
    ! [V_i,V_b] :
      ( ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),V_b),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)))
          | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),V_i),V_b)) )
       => c_Random_Olog(V_b,V_i) = c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral) )
      & ( ~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),V_b),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)))
            | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),V_i),V_b)) )
       => c_Random_Olog(V_b,V_i) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),c_Random_Olog(V_b,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),V_i),V_b))) ) ) )).

fof(fact_minus__shift__def,axiom,(
    ! [V_r,V_l,V_k] :
      ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),V_k),V_l))
       => c_Random_Ominus__shift(V_r,V_k,V_l) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),V_r),V_k)),V_l) )
      & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),V_k),V_l))
       => c_Random_Ominus__shift(V_r,V_k,V_l) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Code__Numeral_Ocode__numeral),V_k),V_l) ) ) )).

fof(fact_inc__shift__def,axiom,(
    ! [V_k,V_v] :
      ( ( V_v = V_k
       => c_Random_Oinc__shift(V_v,V_k) = c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral) )
      & ( V_v != V_k
       => c_Random_Oinc__shift(V_v,V_k) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),V_k),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)) ) ) )).

fof(fact_mod__div__decomp,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Divides_Osemiring__div(T_a)
     => V_a = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_a),V_b)),V_b)),c_Divides_Odiv__class_Omod(T_a,V_a,V_b)) ) )).

fof(fact_minus__Min__eq__Max,axiom,(
    ! [V_S_2,T_b] :
      ( class_Groups_Olinordered__ab__group__add(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_S_2))
       => ( V_S_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
         => hAPP(c_Groups_Ouminus__class_Ouminus(T_b),c_Big__Operators_Olinorder__class_OMin(T_b,V_S_2)) = c_Big__Operators_Olinorder__class_OMax(T_b,hAPP(c_Set_Oimage(T_b,T_b,c_Groups_Ouminus__class_Ouminus(T_b)),V_S_2)) ) ) ) )).

fof(fact_minus__Max__eq__Min,axiom,(
    ! [V_S_2,T_b] :
      ( class_Groups_Olinordered__ab__group__add(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_S_2))
       => ( V_S_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
         => hAPP(c_Groups_Ouminus__class_Ouminus(T_b),c_Big__Operators_Olinorder__class_OMax(T_b,V_S_2)) = c_Big__Operators_Olinorder__class_OMin(T_b,hAPP(c_Set_Oimage(T_b,T_b,c_Groups_Ouminus__class_Ouminus(T_b)),V_S_2)) ) ) ) )).

fof(fact_image__eqI,axiom,(
    ! [T_b,V_A_2,T_c,V_x_2,V_f_2,V_b_2] :
      ( V_b_2 = hAPP(V_f_2,V_x_2)
     => ( hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2))
       => hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),hAPP(c_Set_Oimage(T_c,T_b,V_f_2),V_A_2))) ) ) )).

fof(fact_finite__imageI,axiom,(
    ! [V_h_2,T_c,V_F_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_F_2))
     => hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(c_Set_Oimage(T_b,T_c,V_h_2),V_F_2))) ) )).

fof(fact_transfer__nat__int__set__relations_I5_J,axiom,(
    ! [V_B_2,V_A_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),V_A_2),V_B_2))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),V_A_2)),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),V_B_2))) ) )).

fof(fact_image__mono,axiom,(
    ! [V_f_2,T_c,V_B_2,V_A_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2)),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_B_2))) ) )).

fof(fact_subset__image__iff,axiom,(
    ! [V_A_2,V_f_2,T_c,V_B_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),hAPP(c_Set_Oimage(T_c,T_b,V_f_2),V_A_2)))
    <=> ? [B_AA] :
          ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),B_AA),V_A_2))
          & V_B_2 = hAPP(c_Set_Oimage(T_c,T_b,V_f_2),B_AA) ) ) )).

fof(fact_transfer__nat__int__set__relations_I4_J,axiom,(
    ! [V_B_2,V_A_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),V_A_2),V_B_2))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),V_A_2)),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),V_B_2))) ) )).

fof(fact_image__Pow__surj,axiom,(
    ! [V_B_2,V_A_2,V_f_2,T_b,T_c] :
      ( hAPP(c_Set_Oimage(T_c,T_b,V_f_2),V_A_2) = V_B_2
     => hAPP(c_Set_Oimage(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),c_Set_Oimage(T_c,T_b,V_f_2)),hAPP(c_Set_OPow(T_c),V_A_2)) = hAPP(c_Set_OPow(T_b),V_B_2) ) )).

fof(fact_Nat__Transfer_Otransfer__nat__int__set__functions_I1_J,axiom,(
    ! [V_A_2] : hAPP(c_Finite__Set_Ocard(tc_Nat_Onat),V_A_2) = hAPP(c_Finite__Set_Ocard(tc_Int_Oint),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),V_A_2)) )).

fof(fact_transfer__nat__int__set__relations_I3_J,axiom,(
    ! [V_B_2,V_A_2] :
      ( V_A_2 = V_B_2
    <=> hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),V_A_2) = hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),V_B_2) ) )).

fof(fact_transfer__int__nat__set__return__embed,axiom,(
    ! [V_A_2] : hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),V_A_2)) = V_A_2 )).

fof(fact_image__iff,axiom,(
    ! [V_A_2,V_f_2,T_c,V_z_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(T_b),V_z_2),hAPP(c_Set_Oimage(T_c,T_b,V_f_2),V_A_2)))
    <=> ? [B_x] :
          ( hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2))
          & V_z_2 = hAPP(V_f_2,B_x) ) ) )).

fof(fact_imageI,axiom,(
    ! [V_f_2,T_c,V_A_2,V_x_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
     => hBOOL(hAPP(hAPP(c_member(T_c),hAPP(V_f_2,V_x_2)),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2))) ) )).

fof(fact_rev__image__eqI,axiom,(
    ! [T_c,V_f_2,V_b_2,V_A_2,V_x_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
     => ( V_b_2 = hAPP(V_f_2,V_x_2)
       => hBOOL(hAPP(hAPP(c_member(T_c),V_b_2),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2))) ) ) )).

fof(fact_empty__is__image,axiom,(
    ! [V_A_2,V_f_2,T_c,T_b] :
      ( c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(c_Set_Oimage(T_c,T_b,V_f_2),V_A_2)
    <=> V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) ) )).

fof(fact_image__empty,axiom,(
    ! [V_f_2,T_b,T_c] : hAPP(c_Set_Oimage(T_c,T_b,V_f_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) )).

fof(fact_image__is__empty,axiom,(
    ! [V_A_2,V_f_2,T_b,T_c] :
      ( hAPP(c_Set_Oimage(T_c,T_b,V_f_2),V_A_2) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
    <=> V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) ) )).

fof(fact_image__insert,axiom,(
    ! [V_B_2,V_a_2,V_f_2,T_b,T_c] : hAPP(c_Set_Oimage(T_c,T_b,V_f_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_B_2)) = hAPP(hAPP(c_Set_Oinsert(T_b),hAPP(V_f_2,V_a_2)),hAPP(c_Set_Oimage(T_c,T_b,V_f_2),V_B_2)) )).

fof(fact_Nat__Transfer_Otransfer__nat__int__set__functions_I2_J,axiom,(
    c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool))) )).

fof(fact_Nat__Transfer_Otransfer__int__nat__set__functions_I2_J,axiom,(
    c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) = hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool))) )).

fof(fact_transfer__nat__int__set__relations_I2_J,axiom,(
    ! [V_A_2,V_x_2] :
      ( hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),V_x_2),V_A_2))
    <=> hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_x_2)),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),V_A_2))) ) )).

fof(fact_transfer__nat__int__set__relations_I1_J,axiom,(
    ! [V_A_2] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),V_A_2))
    <=> hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),V_A_2))) ) )).

fof(fact_image__Pow__mono,axiom,(
    ! [V_B_2,V_A_2,V_f_2,T_c,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_c,T_b,V_f_2),V_A_2)),V_B_2))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)),hAPP(c_Set_Oimage(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),c_Set_Oimage(T_c,T_b,V_f_2)),hAPP(c_Set_OPow(T_c),V_A_2))),hAPP(c_Set_OPow(T_b),V_B_2))) ) )).

fof(fact_insert__image,axiom,(
    ! [V_f_2,T_c,V_A_2,V_x_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
     => hAPP(hAPP(c_Set_Oinsert(T_c),hAPP(V_f_2,V_x_2)),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2)) = hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2) ) )).

fof(fact_finite__surj,axiom,(
    ! [V_f_2,V_B_2,T_c,V_A_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_B_2),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2)))
       => hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) ) ) )).

fof(fact_image__diff__subset,axiom,(
    ! [V_B_2,V_A_2,V_f_2,T_c,T_b] : hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_c,T_b,V_f_2),V_A_2)),hAPP(c_Set_Oimage(T_c,T_b,V_f_2),V_B_2))),hAPP(c_Set_Oimage(T_c,T_b,V_f_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)))) )).

fof(fact_card__image__le,axiom,(
    ! [V_f_2,T_c,V_A_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_c),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2))),hAPP(c_Finite__Set_Ocard(T_b),V_A_2))) ) )).

fof(fact_min__max_OSup__fin_Ohom__commute,axiom,(
    ! [V_N_2,V_h_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( ! [B_x,B_y] : hAPP(V_h_2,hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),B_x),B_y)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),hAPP(V_h_2,B_x)),hAPP(V_h_2,B_y))
       => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_N_2))
         => ( V_N_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
           => hAPP(V_h_2,c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),V_N_2)) = c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),hAPP(c_Set_Oimage(T_b,T_b,V_h_2),V_N_2)) ) ) ) ) )).

fof(fact_min__max_OInf__fin_Ohom__commute,axiom,(
    ! [V_N_2,V_h_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( ! [B_x,B_y] : hAPP(V_h_2,hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),B_x),B_y)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),hAPP(V_h_2,B_x)),hAPP(V_h_2,B_y))
       => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_N_2))
         => ( V_N_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
           => hAPP(V_h_2,c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),V_N_2)) = c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),hAPP(c_Set_Oimage(T_b,T_b,V_h_2),V_N_2)) ) ) ) ) )).

fof(fact_Max_Ohom__commute,axiom,(
    ! [V_N_2,V_h_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( ! [B_x,B_y] : hAPP(V_h_2,hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),B_x),B_y)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),hAPP(V_h_2,B_x)),hAPP(V_h_2,B_y))
       => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_N_2))
         => ( V_N_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
           => hAPP(V_h_2,c_Big__Operators_Olinorder__class_OMax(T_b,V_N_2)) = c_Big__Operators_Olinorder__class_OMax(T_b,hAPP(c_Set_Oimage(T_b,T_b,V_h_2),V_N_2)) ) ) ) ) )).

fof(fact_Min_Ohom__commute,axiom,(
    ! [V_N_2,V_h_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( ! [B_x,B_y] : hAPP(V_h_2,hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),B_x),B_y)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),hAPP(V_h_2,B_x)),hAPP(V_h_2,B_y))
       => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_N_2))
         => ( V_N_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
           => hAPP(V_h_2,c_Big__Operators_Olinorder__class_OMin(T_b,V_N_2)) = c_Big__Operators_Olinorder__class_OMin(T_b,hAPP(c_Set_Oimage(T_b,T_b,V_h_2),V_N_2)) ) ) ) ) )).

fof(fact_hom__fold1__commute,axiom,(
    ! [V_N_2,V_h_2,T_b] :
      ( class_Lattices_Oab__semigroup__idem__mult(T_b)
     => ( ! [B_x,B_y] : hAPP(V_h_2,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),B_x),B_y)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(V_h_2,B_x)),hAPP(V_h_2,B_y))
       => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_N_2))
         => ( V_N_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
           => hAPP(V_h_2,hAPP(c_Finite__Set_Ofold1(T_b,c_Groups_Otimes__class_Otimes(T_b)),V_N_2)) = hAPP(c_Finite__Set_Ofold1(T_b,c_Groups_Otimes__class_Otimes(T_b)),hAPP(c_Set_Oimage(T_b,T_b,V_h_2),V_N_2)) ) ) ) ) )).

fof(fact_Cantors__paradox,axiom,(
    ! [V_A_2,T_b] :
      ~ ? [B_f] : hAPP(c_Set_Oimage(T_b,tc_fun(T_b,tc_HOL_Obool),B_f),V_A_2) = hAPP(c_Set_OPow(T_b),V_A_2) )).

fof(fact_finite__nat__set__iff__bounded__le,axiom,(
    ! [V_N_2] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),V_N_2))
    <=> ? [B_m] :
        ! [B_x] :
          ( hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),V_N_2))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B_x),B_m)) ) ) )).

fof(fact_finite__nat__set__iff__bounded,axiom,(
    ! [V_N_2] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),V_N_2))
    <=> ? [B_m] :
        ! [B_x] :
          ( hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),V_N_2))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_x),B_m)) ) ) )).

fof(fact_transfer__int__nat__set__relations_I4_J,axiom,(
    ! [V_B_2,V_A_2] :
      ( c_Nat__Transfer_Onat__set(V_A_2)
     => ( c_Nat__Transfer_Onat__set(V_B_2)
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(tc_Int_Oint,tc_HOL_Obool)),V_A_2),V_B_2))
        <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),V_A_2)),hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),V_B_2))) ) ) ) )).

fof(fact_transfer__int__nat__set__relations_I5_J,axiom,(
    ! [V_B_2,V_A_2] :
      ( c_Nat__Transfer_Onat__set(V_A_2)
     => ( c_Nat__Transfer_Onat__set(V_B_2)
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Int_Oint,tc_HOL_Obool)),V_A_2),V_B_2))
        <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),V_A_2)),hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),V_B_2))) ) ) ) )).

fof(fact_Nat__Transfer_Otransfer__int__nat__set__function__closures_I1_J,axiom,(
    c_Nat__Transfer_Onat__set(c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool))) )).

fof(fact_nat__set__def,axiom,(
    ! [V_S_2] :
      ( c_Nat__Transfer_Onat__set(V_S_2)
    <=> ! [B_x] :
          ( hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_x),V_S_2))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B_x)) ) ) )).

fof(fact_Nat__Transfer_Otransfer__int__nat__set__function__closures_I5_J,axiom,(
    ! [V_C_2] : c_Nat__Transfer_Onat__set(hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),V_C_2)) )).

fof(fact_transfer__int__nat__set__relations_I3_J,axiom,(
    ! [V_B_2,V_A_2] :
      ( c_Nat__Transfer_Onat__set(V_A_2)
     => ( c_Nat__Transfer_Onat__set(V_B_2)
       => ( V_A_2 = V_B_2
        <=> hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),V_A_2) = hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),V_B_2) ) ) ) )).

fof(fact_Nat__Transfer_Otransfer__int__nat__set__function__closures_I6_J,axiom,(
    ! [V_x_2,V_A_2] :
      ( c_Nat__Transfer_Onat__set(V_A_2)
     => ( hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),V_x_2),V_A_2))
       => c_Nat__Transfer_Ois__nat(V_x_2) ) ) )).

fof(fact_Nat__Transfer_Otransfer__nat__int__set__function__closures_I6_J,axiom,(
    ! [V_x_2,V_A_2] :
      ( c_Nat__Transfer_Onat__set(V_A_2)
     => ( hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),V_x_2),V_A_2))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x_2)) ) ) )).

fof(fact_transfer__nat__int__set__return__embed,axiom,(
    ! [V_A_2] :
      ( c_Nat__Transfer_Onat__set(V_A_2)
     => hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),V_A_2)) = V_A_2 ) )).

fof(fact_transfer__int__nat__set__relations_I1_J,axiom,(
    ! [V_A_2] :
      ( c_Nat__Transfer_Onat__set(V_A_2)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),V_A_2))
      <=> hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),V_A_2))) ) ) )).

fof(fact_Nat__Transfer_Otransfer__int__nat__set__functions_I1_J,axiom,(
    ! [V_A_2] :
      ( c_Nat__Transfer_Onat__set(V_A_2)
     => hAPP(c_Finite__Set_Ocard(tc_Int_Oint),V_A_2) = hAPP(c_Finite__Set_Ocard(tc_Nat_Onat),hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),V_A_2)) ) )).

fof(fact_transfer__int__nat__set__relations_I2_J,axiom,(
    ! [V_A_2,V_x_2] :
      ( c_Nat__Transfer_Ois__nat(V_x_2)
     => ( c_Nat__Transfer_Onat__set(V_A_2)
       => ( hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),V_x_2),V_A_2))
        <=> hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),hAPP(c_Int_Onat,V_x_2)),hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),V_A_2))) ) ) ) )).

fof(fact_card__greaterThanLessThan__int,axiom,(
    ! [V_u_2,V_l_2] : hAPP(c_Finite__Set_Ocard(tc_Int_Oint),c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Int_Oint,V_l_2,V_u_2)) = hAPP(c_Int_Onat,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_u_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_l_2),c_Groups_Oone__class_Oone(tc_Int_Oint)))) )).

fof(fact_ord_OgreaterThanAtMost__iff,axiom,(
    ! [V_u_2,V_l_2,V_less_2,V_less__eq_2,V_i_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(T_b),V_i_2),c_SetInterval_Oord_OgreaterThanAtMost(T_b,V_less__eq_2,V_less_2,V_l_2,V_u_2)))
    <=> ( hBOOL(hAPP(hAPP(V_less_2,V_l_2),V_i_2))
        & hBOOL(hAPP(hAPP(V_less__eq_2,V_i_2),V_u_2)) ) ) )).

fof(fact_ord_OatLeastLessThan__iff,axiom,(
    ! [V_u_2,V_l_2,V_less_2,V_less__eq_2,V_i_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(T_b),V_i_2),c_SetInterval_Oord_OatLeastLessThan(T_b,V_less__eq_2,V_less_2,V_l_2,V_u_2)))
    <=> ( hBOOL(hAPP(hAPP(V_less__eq_2,V_l_2),V_i_2))
        & hBOOL(hAPP(hAPP(V_less_2,V_i_2),V_u_2)) ) ) )).

fof(fact_ord_OgreaterThanLessThan__iff,axiom,(
    ! [V_u_2,V_l_2,V_less_2,V_i_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(T_b),V_i_2),c_SetInterval_Oord_OgreaterThanLessThan(T_b,V_less_2,V_l_2,V_u_2)))
    <=> ( hBOOL(hAPP(hAPP(V_less_2,V_l_2),V_i_2))
        & hBOOL(hAPP(hAPP(V_less_2,V_i_2),V_u_2)) ) ) )).

fof(fact_ord_OatLeastAtMost__iff,axiom,(
    ! [V_u_2,V_l_2,V_less__eq_2,V_i_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(T_b),V_i_2),c_SetInterval_Oord_OatLeastAtMost(T_b,V_less__eq_2,V_l_2,V_u_2)))
    <=> ( hBOOL(hAPP(hAPP(V_less__eq_2,V_l_2),V_i_2))
        & hBOOL(hAPP(hAPP(V_less__eq_2,V_i_2),V_u_2)) ) ) )).

fof(fact_finite__greaterThanLessThan__int,axiom,(
    ! [V_u_2,V_l_2] : hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Int_Oint,V_l_2,V_u_2))) )).

fof(fact_greaterThanLessThan__empty,axiom,(
    ! [V_k_2,V_l_2,T_b] :
      ( class_Orderings_Oorder(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_l_2),V_k_2))
       => c_SetInterval_Oord__class_OgreaterThanLessThan(T_b,V_k_2,V_l_2) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) ) ) )).

fof(fact_image__uminus__greaterThanLessThan,axiom,(
    ! [V_y_2,V_x_2,T_b] :
      ( class_Groups_Oordered__ab__group__add(T_b)
     => hAPP(c_Set_Oimage(T_b,T_b,c_Groups_Ouminus__class_Ouminus(T_b)),c_SetInterval_Oord__class_OgreaterThanLessThan(T_b,V_x_2,V_y_2)) = c_SetInterval_Oord__class_OgreaterThanLessThan(T_b,hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_y_2),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_x_2)) ) )).

fof(fact_ord_OatMost__iff,axiom,(
    ! [V_k_2,V_less__eq_2,V_i_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(T_b),V_i_2),c_SetInterval_Oord_OatMost(T_b,V_less__eq_2,V_k_2)))
    <=> hBOOL(hAPP(hAPP(V_less__eq_2,V_i_2),V_k_2)) ) )).

fof(fact_ord_OatLeast__iff,axiom,(
    ! [V_k_2,V_less__eq_2,V_i_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(T_b),V_i_2),c_SetInterval_Oord_OatLeast(T_b,V_less__eq_2,V_k_2)))
    <=> hBOOL(hAPP(hAPP(V_less__eq_2,V_k_2),V_i_2)) ) )).

fof(fact_ord_OlessThan__iff,axiom,(
    ! [V_k_2,V_less_2,V_i_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(T_b),V_i_2),c_SetInterval_Oord_OlessThan(T_b,V_less_2,V_k_2)))
    <=> hBOOL(hAPP(hAPP(V_less_2,V_i_2),V_k_2)) ) )).

fof(fact_ord_OgreaterThan__iff,axiom,(
    ! [V_k_2,V_less_2,V_i_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(T_b),V_i_2),c_SetInterval_Oord_OgreaterThan(T_b,V_less_2,V_k_2)))
    <=> hBOOL(hAPP(hAPP(V_less_2,V_k_2),V_i_2)) ) )).

fof(fact_fun__upd__image,axiom,(
    ! [V_y_2,V_f_2,T_c,V_A_2,V_x_2,T_b] :
      ( ( hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
       => hAPP(c_Set_Oimage(T_b,T_c,c_Fun_Ofun__upd(T_b,T_c,V_f_2,V_x_2,V_y_2)),V_A_2) = hAPP(hAPP(c_Set_Oinsert(T_c),V_y_2),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))) )
      & ( ~ hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
       => hAPP(c_Set_Oimage(T_b,T_c,c_Fun_Ofun__upd(T_b,T_c,V_f_2,V_x_2,V_y_2)),V_A_2) = hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2) ) ) )).

fof(fact_fun__upd__triv,axiom,(
    ! [V_x_2,V_f_2,T_c,T_b] : c_Fun_Ofun__upd(T_b,T_c,V_f_2,V_x_2,hAPP(V_f_2,V_x_2)) = V_f_2 )).

fof(fact_finite__greaterThanLessThan,axiom,(
    ! [V_u_2,V_l_2] : hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Nat_Onat,V_l_2,V_u_2))) )).

fof(fact_fun__upd__idem__iff,axiom,(
    ! [V_y_2,V_x_2,V_f_2,T_c,T_b] :
      ( c_Fun_Ofun__upd(T_b,T_c,V_f_2,V_x_2,V_y_2) = V_f_2
    <=> hAPP(V_f_2,V_x_2) = V_y_2 ) )).

fof(fact_fun__upd__upd,axiom,(
    ! [V_z_2,V_y_2,V_x_2,V_f_2,T_c,T_b] : c_Fun_Ofun__upd(T_b,T_c,c_Fun_Ofun__upd(T_b,T_c,V_f_2,V_x_2,V_y_2),V_x_2,V_z_2) = c_Fun_Ofun__upd(T_b,T_c,V_f_2,V_x_2,V_z_2) )).

fof(fact_fun__upd__same,axiom,(
    ! [V_y_2,V_x_2,V_f_2,T_b,T_c] : hAPP(c_Fun_Ofun__upd(T_c,T_b,V_f_2,V_x_2,V_y_2),V_x_2) = V_y_2 )).

fof(fact_fun__upd__apply,axiom,(
    ! [V_y_2,V_f_2,T_c,T_b,V_x_2,V_z_2] :
      ( ( V_z_2 = V_x_2
       => hAPP(c_Fun_Ofun__upd(T_b,T_c,V_f_2,V_x_2,V_y_2),V_z_2) = V_y_2 )
      & ( V_z_2 != V_x_2
       => hAPP(c_Fun_Ofun__upd(T_b,T_c,V_f_2,V_x_2,V_y_2),V_z_2) = hAPP(V_f_2,V_z_2) ) ) )).

fof(fact_fun__upd__twist,axiom,(
    ! [V_d_2,V_b_2,V_m_2,T_c,T_b,V_c_2,V_a_2] :
      ( V_a_2 != V_c_2
     => c_Fun_Ofun__upd(T_b,T_c,c_Fun_Ofun__upd(T_b,T_c,V_m_2,V_a_2,V_b_2),V_c_2,V_d_2) = c_Fun_Ofun__upd(T_b,T_c,c_Fun_Ofun__upd(T_b,T_c,V_m_2,V_c_2,V_d_2),V_a_2,V_b_2) ) )).

fof(fact_fun__upd__other,axiom,(
    ! [V_y_2,V_f_2,T_c,T_b,V_x_2,V_z_2] :
      ( V_z_2 != V_x_2
     => hAPP(c_Fun_Ofun__upd(T_b,T_c,V_f_2,V_x_2,V_y_2),V_z_2) = hAPP(V_f_2,V_z_2) ) )).

fof(fact_fun__upd__idem,axiom,(
    ! [T_b,T_c,V_y_2,V_x_2,V_f_2] :
      ( hAPP(V_f_2,V_x_2) = V_y_2
     => c_Fun_Ofun__upd(T_c,T_b,V_f_2,V_x_2,V_y_2) = V_f_2 ) )).

fof(fact_card__greaterThanLessThan,axiom,(
    ! [V_u_2,V_l_2] : hAPP(c_Finite__Set_Ocard(tc_Nat_Onat),c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Nat_Onat,V_l_2,V_u_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_u_2),hAPP(c_Nat_OSuc,V_l_2)) )).

fof(fact_folding__one__idem_Ohom__commute,axiom,(
    ! [V_N_2,V_h_2,V_F_2,V_f_2,T_b] :
      ( c_Finite__Set_Ofolding__one__idem(T_b,V_f_2,V_F_2)
     => ( ! [B_x,B_y] : hAPP(V_h_2,hAPP(hAPP(V_f_2,B_x),B_y)) = hAPP(hAPP(V_f_2,hAPP(V_h_2,B_x)),hAPP(V_h_2,B_y))
       => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_N_2))
         => ( V_N_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
           => hAPP(V_h_2,hAPP(V_F_2,V_N_2)) = hAPP(V_F_2,hAPP(c_Set_Oimage(T_b,T_b,V_h_2),V_N_2)) ) ) ) ) )).

fof(fact_card__atLeastAtMost__int,axiom,(
    ! [V_u_2,V_l_2] : hAPP(c_Finite__Set_Ocard(tc_Int_Oint),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,V_l_2,V_u_2)) = hAPP(c_Int_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_u_2),V_l_2)),c_Groups_Oone__class_Oone(tc_Int_Oint))) )).

fof(fact_minus__code__numeral__code,axiom,(
    ! [V_m,V_n] : hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Code__Numeral_Ocode__numeral),V_n),V_m) = hAPP(hAPP(c_Code__Numeral_Osubtract__code__numeral,V_n),V_m) )).

fof(fact_subtract__code__numeral__def,axiom,(
    c_Code__Numeral_Osubtract__code__numeral = c_Groups_Ominus__class_Ominus(tc_Code__Numeral_Ocode__numeral) )).

fof(fact_negDivAlg__eqn__1__number__of,axiom,(
    ! [V_w] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)))
     => ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w))))
         => c_Divides_OnegDivAlg(c_Groups_Oone__class_Oone(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)) = hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w))) )
        & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w))))
         => c_Divides_OnegDivAlg(c_Groups_Oone__class_Oone(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)) = hAPP(c_Divides_Oadjust(hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)),c_Divides_OnegDivAlg(c_Groups_Oone__class_Oone(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)))) ) ) ) )).

fof(fact_finite__atLeastAtMost__int,axiom,(
    ! [V_u_2,V_l_2] : hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,V_l_2,V_u_2))) )).

fof(fact_SetInterval_Otransfer__int__nat__set__function__closures,axiom,(
    ! [V_y_2,V_x_2] :
      ( c_Nat__Transfer_Ois__nat(V_x_2)
     => c_Nat__Transfer_Onat__set(c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,V_x_2,V_y_2)) ) )).

fof(fact_negDivAlg__div__mod,axiom,(
    ! [V_l,V_k] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_l))
       => c_Divides_OnegDivAlg(V_k,V_l) = hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_k),V_l)),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_k,V_l)) ) ) )).

fof(fact_atLeastatMost__empty__iff,axiom,(
    ! [V_b_2,V_a_2,T_b] :
      ( class_Orderings_Oorder(T_b)
     => ( c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_a_2,V_b_2) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
      <=> ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),V_b_2)) ) ) )).

fof(fact_atLeastatMost__empty__iff2,axiom,(
    ! [V_b_2,V_a_2,T_b] :
      ( class_Orderings_Oorder(T_b)
     => ( c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_a_2,V_b_2)
      <=> ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),V_b_2)) ) ) )).

fof(fact_atLeastatMost__empty,axiom,(
    ! [V_a_2,V_b_2,T_b] :
      ( class_Orderings_Oorder(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_b_2),V_a_2))
       => c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_a_2,V_b_2) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) ) ) )).

fof(fact_negDivAlg__minus1,axiom,(
    ! [V_b] : c_Divides_OnegDivAlg(hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin),V_b) = hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_b),c_Groups_Oone__class_Oone(tc_Int_Oint))) )).

fof(fact_atLeastatMost__subset__iff,axiom,(
    ! [V_d_2,V_c_2,V_b_2,V_a_2,T_b] :
      ( class_Orderings_Oorder(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_a_2,V_b_2)),c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_c_2,V_d_2)))
      <=> ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),V_b_2))
          | ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_c_2),V_a_2))
            & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_b_2),V_d_2)) ) ) ) ) )).

fof(fact_atLeastAtMost__singleton_H,axiom,(
    ! [V_b_2,V_a_2,T_b] :
      ( class_Orderings_Oorder(T_b)
     => ( V_a_2 = V_b_2
       => c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_a_2,V_b_2) = hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) ) ) )).

fof(fact_atLeastAtMost__singleton__iff,axiom,(
    ! [V_c_2,V_b_2,V_a_2,T_b] :
      ( class_Orderings_Oorder(T_b)
     => ( c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_a_2,V_b_2) = hAPP(hAPP(c_Set_Oinsert(T_b),V_c_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))
      <=> ( V_a_2 = V_b_2
          & V_b_2 = V_c_2 ) ) ) )).

fof(fact_atLeastAtMost__singleton,axiom,(
    ! [V_a_2,T_b] :
      ( class_Orderings_Oorder(T_b)
     => c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_a_2,V_a_2) = hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) ) )).

fof(fact_image__uminus__atLeastAtMost,axiom,(
    ! [V_y_2,V_x_2,T_b] :
      ( class_Groups_Oordered__ab__group__add(T_b)
     => hAPP(c_Set_Oimage(T_b,T_b,c_Groups_Ouminus__class_Ouminus(T_b)),c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_x_2,V_y_2)) = c_SetInterval_Oord__class_OatLeastAtMost(T_b,hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_y_2),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_x_2)) ) )).

fof(fact_SetInterval_Otransfer__nat__int__set__function__closures,axiom,(
    ! [V_y_2,V_x_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x_2))
     => c_Nat__Transfer_Onat__set(c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,V_x_2,V_y_2)) ) )).

fof(fact_atLeastatMost__psubset__iff,axiom,(
    ! [V_d_2,V_c_2,V_b_2,V_a_2,T_b] :
      ( class_Orderings_Oorder(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_b,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_a_2,V_b_2)),c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_c_2,V_d_2)))
      <=> ( ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),V_b_2))
            | ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_c_2),V_a_2))
              & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_b_2),V_d_2))
              & ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),V_a_2))
                | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_b_2),V_d_2)) ) ) )
          & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_c_2),V_d_2)) ) ) ) )).

fof(fact_negDivAlg_Osimps,axiom,(
    ! [V_b,V_a] :
      ( ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a),V_b)))
          | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_b),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) )
       => c_Divides_OnegDivAlg(V_a,V_b) = hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a),V_b)) )
      & ( ~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a),V_b)))
            | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_b),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) )
       => c_Divides_OnegDivAlg(V_a,V_b) = hAPP(c_Divides_Oadjust(V_b),c_Divides_OnegDivAlg(V_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_b))) ) ) )).

fof(fact_negDivAlg__eqn,axiom,(
    ! [V_a,V_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b))
     => ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a),V_b)))
         => c_Divides_OnegDivAlg(V_a,V_b) = hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a),V_b)) )
        & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a),V_b)))
         => c_Divides_OnegDivAlg(V_a,V_b) = hAPP(c_Divides_Oadjust(V_b),c_Divides_OnegDivAlg(V_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_b))) ) ) ) )).

fof(fact_negDivAlg__eqn__number__of,axiom,(
    ! [V_w,V_v] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)))
     => ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))))
         => c_Divides_OnegDivAlg(hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)) = hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))) )
        & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))))
         => c_Divides_OnegDivAlg(hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)) = hAPP(c_Divides_Oadjust(hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)),c_Divides_OnegDivAlg(hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)))) ) ) ) )).

fof(fact_simp__from__to,axiom,(
    ! [V_i_2,V_j_2] :
      ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_j_2),V_i_2))
       => c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,V_i_2,V_j_2) = c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) )
      & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_j_2),V_i_2))
       => c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,V_i_2,V_j_2) = hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),V_i_2),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_i_2),c_Groups_Oone__class_Oone(tc_Int_Oint)),V_j_2)) ) ) )).

fof(fact_posDivAlg__eqn__1__number__of,axiom,(
    ! [V_w] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)))
     => ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)))
         => c_Divides_OposDivAlg(c_Groups_Oone__class_Oone(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)) = hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Groups_Oone__class_Oone(tc_Int_Oint)) )
        & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)))
         => c_Divides_OposDivAlg(c_Groups_Oone__class_Oone(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)) = hAPP(c_Divides_Oadjust(hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)),c_Divides_OposDivAlg(c_Groups_Oone__class_Oone(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)))) ) ) ) )).

fof(fact_posDivAlg_Osimps,axiom,(
    ! [V_b,V_a] :
      ( ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_a),V_b))
          | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_b),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) )
       => c_Divides_OposDivAlg(V_a,V_b) = hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a) )
      & ( ~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_a),V_b))
            | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_b),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) )
       => c_Divides_OposDivAlg(V_a,V_b) = hAPP(c_Divides_Oadjust(V_b),c_Divides_OposDivAlg(V_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_b))) ) ) )).

fof(fact_in__rel__def,axiom,(
    ! [V_y_2,V_x_2,V_R_2,T_c,T_b] :
      ( hBOOL(hAPP(hAPP(c_FunDef_Oin__rel(T_b,T_c,V_R_2),V_x_2),V_y_2))
    <=> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_x_2),V_y_2)),V_R_2)) ) )).

fof(fact_posDivAlg__eqn,axiom,(
    ! [V_a,V_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b))
     => ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_a),V_b))
         => c_Divides_OposDivAlg(V_a,V_b) = hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a) )
        & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_a),V_b))
         => c_Divides_OposDivAlg(V_a,V_b) = hAPP(c_Divides_Oadjust(V_b),c_Divides_OposDivAlg(V_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_b))) ) ) ) )).

fof(fact_posDivAlg__eqn__number__of,axiom,(
    ! [V_w,V_v] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)))
     => ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)))
         => c_Divides_OposDivAlg(hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)) = hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)) )
        & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)))
         => c_Divides_OposDivAlg(hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)) = hAPP(c_Divides_Oadjust(hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)),c_Divides_OposDivAlg(hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)))) ) ) ) )).

fof(fact_finite__atLeastAtMost,axiom,(
    ! [V_u_2,V_l_2] : hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_l_2,V_u_2))) )).

fof(fact_image__Suc__atLeastAtMost,axiom,(
    ! [V_j_2,V_i_2] : hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Nat_Onat,c_Nat_OSuc),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_i_2,V_j_2)) = c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_i_2),hAPP(c_Nat_OSuc,V_j_2)) )).

fof(fact_setsum__shift__lb__Suc0__0,axiom,(
    ! [V_k_2,V_f_2] :
      ( hAPP(V_f_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
     => hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),V_f_2),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_k_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),V_f_2),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_k_2)) ) )).

fof(fact_atLeastAtMostSuc__conv,axiom,(
    ! [V_n_2,V_m_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),hAPP(c_Nat_OSuc,V_n_2)))
     => c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_m_2,hAPP(c_Nat_OSuc,V_n_2)) = hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_n_2)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_m_2,V_n_2)) ) )).

fof(fact_setsum__head__Suc,axiom,(
    ! [V_f_2,V_n_2,V_m_2,T_b] :
      ( class_Groups_Ocomm__monoid__add(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2))
       => hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_m_2,V_n_2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(V_f_2,V_m_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m_2),V_n_2))) ) ) )).

fof(fact_posDivAlg__0,axiom,(
    ! [V_b] : c_Divides_OposDivAlg(c_Groups_Ozero__class_Ozero(tc_Int_Oint),V_b) = hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Groups_Ozero__class_Ozero(tc_Int_Oint)) )).

fof(fact_card__atLeastAtMost,axiom,(
    ! [V_u_2,V_l_2] : hAPP(c_Finite__Set_Ocard(tc_Nat_Onat),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_l_2,V_u_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_u_2)),V_l_2) )).

fof(fact_setsum__cl__ivl__Suc,axiom,(
    ! [V_f_2,V_m_2,V_n_2,T_b] :
      ( class_Groups_Ocomm__monoid__add(T_b)
     => ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_n_2)),V_m_2))
         => hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_m_2,hAPP(c_Nat_OSuc,V_n_2))) = c_Groups_Ozero__class_Ozero(T_b) )
        & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_n_2)),V_m_2))
         => hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_m_2,hAPP(c_Nat_OSuc,V_n_2))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_m_2,V_n_2))),hAPP(V_f_2,hAPP(c_Nat_OSuc,V_n_2))) ) ) ) )).

fof(fact_setsum__ub__add__nat,axiom,(
    ! [V_p_2,V_f_2,V_n_2,V_m_2,T_b] :
      ( class_Groups_Ocomm__monoid__add(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n_2),c_Groups_Oone__class_Oone(tc_Nat_Onat))))
       => hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_m_2,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n_2),V_p_2))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_m_2,V_n_2))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n_2),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n_2),V_p_2)))) ) ) )).

fof(fact_SetInterval_Otransfer__nat__int__set__functions_I2_J,axiom,(
    ! [V_n_2,V_m_2] : c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_m_2,V_n_2) = hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m_2),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n_2))) )).

fof(fact_SetInterval_Otransfer__int__nat__set__functions,axiom,(
    ! [V_n_2,V_m_2] :
      ( c_Nat__Transfer_Ois__nat(V_m_2)
     => ( c_Nat__Transfer_Ois__nat(V_n_2)
       => c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,V_m_2,V_n_2) = hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(c_Int_Onat,V_m_2),hAPP(c_Int_Onat,V_n_2))) ) ) )).

fof(fact_posDivAlg__div__mod,axiom,(
    ! [V_l,V_k] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_k))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_l))
       => c_Divides_OposDivAlg(V_k,V_l) = hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_k),V_l)),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_k,V_l)) ) ) )).

fof(fact_div__mod__code__numeral__def,axiom,(
    ! [V_m,V_n] : c_Code__Numeral_Odiv__mod__code__numeral(V_n,V_m) = hAPP(hAPP(c_Product__Type_OPair(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),V_n),V_m)),c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,V_n,V_m)) )).

fof(fact_Nitpick_Orefl_H__def,axiom,(
    ! [V_r_2,T_b] :
      ( c_Nitpick_Orefl_H(T_b,V_r_2)
    <=> ! [B_x] : hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),B_x),B_x)),V_r_2)) ) )).

fof(fact_divmod__nat__step,axiom,(
    ! [V_m,V_n] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n),V_m))
       => c_Divides_Odivmod__nat(V_m,V_n) = hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),hAPP(c_Nat_OSuc,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)),V_n))),c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n),V_n)) ) ) )).

fof(fact_aset_I6_J,axiom,(
    ! [V_A_2,V_ta_2,V_D_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_D_2))
     => ( hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_ta_2),c_Groups_Oone__class_Oone(tc_Int_Oint))),V_A_2))
       => ! [B_x] :
            ( ! [B_xa] :
                ( hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xa),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),V_D_2)))
               => ! [B_xb] :
                    ( hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xb),V_A_2))
                   => B_x != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B_xb),B_xa) ) )
           => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B_x),V_ta_2))
             => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B_x),V_D_2)),V_ta_2)) ) ) ) ) )).

fof(fact_bset_I8_J,axiom,(
    ! [V_B_2,V_ta_2,V_D_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_D_2))
     => ( hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_ta_2),c_Groups_Oone__class_Oone(tc_Int_Oint))),V_B_2))
       => ! [B_x] :
            ( ! [B_xa] :
                ( hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xa),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),V_D_2)))
               => ! [B_xb] :
                    ( hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xb),V_B_2))
                   => B_x != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B_xb),B_xa) ) )
           => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_ta_2),B_x))
             => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_ta_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B_x),V_D_2))) ) ) ) ) )).

fof(fact_divmod__nat__zero,axiom,(
    ! [V_m] : c_Divides_Odivmod__nat(V_m,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_m) )).

fof(fact_divmod__nat__base,axiom,(
    ! [V_n,V_m] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n))
     => c_Divides_Odivmod__nat(V_m,V_n) = hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_m) ) )).

fof(fact_divmod__nat__div__mod,axiom,(
    ! [V_n,V_m] : c_Divides_Odivmod__nat(V_m,V_n) = hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),V_n)),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n)) )).

fof(fact_all__nat__less,axiom,(
    ! [V_P_2,V_n_2] :
      ( ! [B_m] :
          ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B_m),V_n_2))
         => hBOOL(hAPP(V_P_2,B_m)) )
    <=> ! [B_x] :
          ( hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
         => hBOOL(hAPP(V_P_2,B_x)) ) ) )).

fof(fact_ex__nat__less,axiom,(
    ! [V_P_2,V_n_2] :
      ( ? [B_m] :
          ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B_m),V_n_2))
          & hBOOL(hAPP(V_P_2,B_m)) )
    <=> ? [B_x] :
          ( hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)))
          & hBOOL(hAPP(V_P_2,B_x)) ) ) )).

fof(fact_divmod__nat__rel__mult1__eq,axiom,(
    ! [V_a,V_r,V_q,V_c,V_b] :
      ( hBOOL(hAPP(c_Divides_Odivmod__nat__rel(V_b,V_c),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_q),V_r)))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_c))
       => hBOOL(hAPP(c_Divides_Odivmod__nat__rel(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_a),V_b),V_c),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_a),V_q)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_a),V_r)),V_c))),c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_a),V_r),V_c)))) ) ) )).

fof(fact_divmod__nat__rel__mult2__eq,axiom,(
    ! [V_c,V_r,V_q,V_b,V_a] :
      ( hBOOL(hAPP(c_Divides_Odivmod__nat__rel(V_a,V_b),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_q),V_r)))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_b))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_c))
         => hBOOL(hAPP(c_Divides_Odivmod__nat__rel(V_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_b),V_c)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_q),V_c)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_b),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_q,V_c))),V_r)))) ) ) ) )).

fof(fact_negDivAlg_Opsimps,axiom,(
    ! [V_b_2,V_a_2] :
      ( hBOOL(hAPP(c_Wellfounded_Oaccp(tc_prod(tc_Int_Oint,tc_Int_Oint),c_Divides_OnegDivAlg__rel),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_a_2),V_b_2)))
     => ( ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a_2),V_b_2)))
            | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_b_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) )
         => c_Divides_OnegDivAlg(V_a_2,V_b_2) = hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a_2),V_b_2)) )
        & ( ~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a_2),V_b_2)))
              | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_b_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) )
         => c_Divides_OnegDivAlg(V_a_2,V_b_2) = hAPP(c_Divides_Oadjust(V_b_2),c_Divides_OnegDivAlg(V_a_2,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_b_2))) ) ) ) )).

fof(fact_accp_Osimps,axiom,(
    ! [V_a_2,V_r_2,T_b] :
      ( hBOOL(hAPP(c_Wellfounded_Oaccp(T_b,V_r_2),V_a_2))
    <=> ! [B_x] :
          ( hBOOL(hAPP(hAPP(V_r_2,B_x),V_a_2))
         => hBOOL(hAPP(c_Wellfounded_Oaccp(T_b,V_r_2),B_x)) ) ) )).

fof(fact_accp_Oequations,axiom,(
    ! [V_a_2,V_r_2,T_b] :
      ( hBOOL(hAPP(c_Wellfounded_Oaccp(T_b,V_r_2),V_a_2))
    <=> ! [B_x] :
          ( hBOOL(hAPP(hAPP(V_r_2,B_x),V_a_2))
         => hBOOL(hAPP(c_Wellfounded_Oaccp(T_b,V_r_2),B_x)) ) ) )).

fof(fact_accp__downward,axiom,(
    ! [V_a_2,V_b_2,V_r_2,T_b] :
      ( hBOOL(hAPP(c_Wellfounded_Oaccp(T_b,V_r_2),V_b_2))
     => ( hBOOL(hAPP(hAPP(V_r_2,V_a_2),V_b_2))
       => hBOOL(hAPP(c_Wellfounded_Oaccp(T_b,V_r_2),V_a_2)) ) ) )).

fof(fact_divmod__nat__rel__unique,axiom,(
    ! [V_qr_H,V_qr,V_n,V_m] :
      ( hBOOL(hAPP(c_Divides_Odivmod__nat__rel(V_m,V_n),V_qr))
     => ( hBOOL(hAPP(c_Divides_Odivmod__nat__rel(V_m,V_n),V_qr_H))
       => V_qr = V_qr_H ) ) )).

fof(fact_accp__subset,axiom,(
    ! [V_R2_2,V_R1_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_fun(T_b,tc_HOL_Obool))),V_R1_2),V_R2_2))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),c_Wellfounded_Oaccp(T_b,V_R2_2)),c_Wellfounded_Oaccp(T_b,V_R1_2))) ) )).

fof(fact_divmod__nat__rel__divmod__nat,axiom,(
    ! [V_n,V_m] : hBOOL(hAPP(c_Divides_Odivmod__nat__rel(V_m,V_n),c_Divides_Odivmod__nat(V_m,V_n))) )).

fof(fact_divmod__nat__eq,axiom,(
    ! [V_qr,V_n,V_m] :
      ( hBOOL(hAPP(c_Divides_Odivmod__nat__rel(V_m,V_n),V_qr))
     => c_Divides_Odivmod__nat(V_m,V_n) = V_qr ) )).

fof(fact_mod__eq,axiom,(
    ! [V_r,V_q,V_n,V_m] :
      ( hBOOL(hAPP(c_Divides_Odivmod__nat__rel(V_m,V_n),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_q),V_r)))
     => c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n) = V_r ) )).

fof(fact_div__eq,axiom,(
    ! [V_r,V_q,V_n,V_m] :
      ( hBOOL(hAPP(c_Divides_Odivmod__nat__rel(V_m,V_n),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_q),V_r)))
     => hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),V_n) = V_q ) )).

fof(fact_divmod__nat__rel,axiom,(
    ! [V_n,V_m] : hBOOL(hAPP(c_Divides_Odivmod__nat__rel(V_m,V_n),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),V_n)),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n)))) )).

fof(fact_divmod__nat__rel__add1__eq,axiom,(
    ! [V_br,V_bq,V_b,V_ar,V_aq,V_c,V_a] :
      ( hBOOL(hAPP(c_Divides_Odivmod__nat__rel(V_a,V_c),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_aq),V_ar)))
     => ( hBOOL(hAPP(c_Divides_Odivmod__nat__rel(V_b,V_c),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_bq),V_br)))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_c))
         => hBOOL(hAPP(c_Divides_Odivmod__nat__rel(hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_a),V_b),V_c),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_aq),V_bq)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_ar),V_br)),V_c))),c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_ar),V_br),V_c)))) ) ) ) )).

fof(fact_posDivAlg_Opsimps,axiom,(
    ! [V_b_2,V_a_2] :
      ( hBOOL(hAPP(c_Wellfounded_Oaccp(tc_prod(tc_Int_Oint,tc_Int_Oint),c_Divides_OposDivAlg__rel),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_a_2),V_b_2)))
     => ( ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_a_2),V_b_2))
            | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_b_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) )
         => c_Divides_OposDivAlg(V_a_2,V_b_2) = hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a_2) )
        & ( ~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_a_2),V_b_2))
              | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_b_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) )
         => c_Divides_OposDivAlg(V_a_2,V_b_2) = hAPP(c_Divides_Oadjust(V_b_2),c_Divides_OposDivAlg(V_a_2,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_b_2))) ) ) ) )).

fof(fact_negDivAlg_Opinduct,axiom,(
    ! [V_P_2,V_a1_2,V_a0_2] :
      ( hBOOL(hAPP(c_Wellfounded_Oaccp(tc_prod(tc_Int_Oint,tc_Int_Oint),c_Divides_OnegDivAlg__rel),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_a0_2),V_a1_2)))
     => ( ! [B_a,B_b] :
            ( hBOOL(hAPP(c_Wellfounded_Oaccp(tc_prod(tc_Int_Oint,tc_Int_Oint),c_Divides_OnegDivAlg__rel),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),B_a),B_b)))
           => ( ( ~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B_a),B_b)))
                    | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B_b),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) )
               => hBOOL(hAPP(hAPP(V_P_2,B_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),B_b))) )
             => hBOOL(hAPP(hAPP(V_P_2,B_a),B_b)) ) )
       => hBOOL(hAPP(hAPP(V_P_2,V_a0_2),V_a1_2)) ) ) )).

fof(fact_pdivmod__posDivAlg,axiom,(
    ! [V_k,V_l] :
      ( ( V_l = c_Groups_Ozero__class_Ozero(tc_Int_Oint)
       => c_Divides_Opdivmod(V_k,V_l) = hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_k)) )
      & ( V_l != c_Groups_Ozero__class_Ozero(tc_Int_Oint)
       => c_Divides_Opdivmod(V_k,V_l) = c_Divides_OposDivAlg(hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_k),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_l)) ) ) )).

fof(fact_pdivmod__def,axiom,(
    ! [V_l,V_k] : c_Divides_Opdivmod(V_k,V_l) = hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_k)),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_l))),c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_k),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_l))) )).

fof(fact_zmult2__lemma,axiom,(
    ! [V_c,V_r,V_q,V_b,V_a] :
      ( hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,V_b),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_q),V_r)))
     => ( V_b != c_Groups_Ozero__class_Ozero(tc_Int_Oint)
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_c))
         => hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),V_c)),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_q),V_c)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_q,V_c))),V_r)))) ) ) ) )).

fof(fact_rev__predicate2D,axiom,(
    ! [V_Q_2,T_c,T_b,V_y_2,V_x_2,V_P_2] :
      ( hBOOL(hAPP(hAPP(V_P_2,V_x_2),V_y_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_fun(T_c,tc_HOL_Obool))),V_P_2),V_Q_2))
       => hBOOL(hAPP(hAPP(V_Q_2,V_x_2),V_y_2)) ) ) )).

fof(fact_predicate2D,axiom,(
    ! [V_y_2,V_x_2,V_Q_2,V_P_2,T_c,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_fun(T_c,tc_HOL_Obool))),V_P_2),V_Q_2))
     => ( hBOOL(hAPP(hAPP(V_P_2,V_x_2),V_y_2))
       => hBOOL(hAPP(hAPP(V_Q_2,V_x_2),V_y_2)) ) ) )).

fof(fact_divmod__int__rel__0,axiom,(
    ! [V_b] :
      ( V_b != c_Groups_Ozero__class_Ozero(tc_Int_Oint)
     => hBOOL(hAPP(c_Divides_Odivmod__int__rel(c_Groups_Ozero__class_Ozero(tc_Int_Oint),V_b),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))) ) )).

fof(fact_self__remainder,axiom,(
    ! [V_r,V_q,V_a] :
      ( hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,V_a),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_q),V_r)))
     => ( V_a != c_Groups_Ozero__class_Ozero(tc_Int_Oint)
       => V_r = c_Groups_Ozero__class_Ozero(tc_Int_Oint) ) ) )).

fof(fact_unique__quotient,axiom,(
    ! [V_r_H,V_q_H,V_r,V_q,V_b,V_a] :
      ( hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,V_b),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_q),V_r)))
     => ( hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,V_b),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_q_H),V_r_H)))
       => ( V_b != c_Groups_Ozero__class_Ozero(tc_Int_Oint)
         => V_q = V_q_H ) ) ) )).

fof(fact_unique__remainder,axiom,(
    ! [V_r_H,V_q_H,V_r,V_q,V_b,V_a] :
      ( hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,V_b),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_q),V_r)))
     => ( hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,V_b),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_q_H),V_r_H)))
       => ( V_b != c_Groups_Ozero__class_Ozero(tc_Int_Oint)
         => V_r = V_r_H ) ) ) )).

fof(fact_self__quotient,axiom,(
    ! [V_r,V_q,V_a] :
      ( hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,V_a),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_q),V_r)))
     => ( V_a != c_Groups_Ozero__class_Ozero(tc_Int_Oint)
       => V_q = c_Groups_Oone__class_Oone(tc_Int_Oint) ) ) )).

fof(fact_divmod__int__rel__mod,axiom,(
    ! [V_r,V_q,V_b,V_a] :
      ( hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,V_b),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_q),V_r)))
     => ( V_b != c_Groups_Ozero__class_Ozero(tc_Int_Oint)
       => c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b) = V_r ) ) )).

fof(fact_divmod__int__rel__div,axiom,(
    ! [V_r,V_q,V_b,V_a] :
      ( hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,V_b),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_q),V_r)))
     => ( V_b != c_Groups_Ozero__class_Ozero(tc_Int_Oint)
       => hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b) = V_q ) ) )).

fof(fact_negDivAlg__correct,axiom,(
    ! [V_b,V_a] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_a),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b))
       => hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,V_b),c_Divides_OnegDivAlg(V_a,V_b))) ) ) )).

fof(fact_divmod__int__rel__div__mod,axiom,(
    ! [V_a,V_b] :
      ( V_b != c_Groups_Ozero__class_Ozero(tc_Int_Oint)
     => hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,V_b),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b)),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b)))) ) )).

fof(fact_posDivAlg__correct,axiom,(
    ! [V_b,V_a] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b))
       => hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,V_b),c_Divides_OposDivAlg(V_a,V_b))) ) ) )).

fof(fact_zadd1__lemma,axiom,(
    ! [V_br,V_bq,V_b,V_ar,V_aq,V_c,V_a] :
      ( hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,V_c),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_aq),V_ar)))
     => ( hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_b,V_c),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_bq),V_br)))
       => ( V_c != c_Groups_Ozero__class_Ozero(tc_Int_Oint)
         => hBOOL(hAPP(c_Divides_Odivmod__int__rel(hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a),V_b),V_c),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_aq),V_bq)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_ar),V_br)),V_c))),c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_ar),V_br),V_c)))) ) ) ) )).

fof(fact_zminus1__lemma,axiom,(
    ! [V_r_2,V_q_2,V_b_2,V_a_2] :
      ( hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a_2,V_b_2),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_q_2),V_r_2)))
     => hBOOL(hAPP(c_Divides_Odivmod__int__rel(hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a_2),V_b_2),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(hAPP(c_If(tc_Int_Oint),hAPP(hAPP(c_fequal,V_r_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_q_2)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_q_2)),c_Groups_Oone__class_Oone(tc_Int_Oint)))),hAPP(hAPP(hAPP(c_If(tc_Int_Oint),hAPP(hAPP(c_fequal,V_r_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_b_2),V_r_2))))) ) )).

fof(fact_divmod__int__relI,axiom,(
    ! [V_r,V_q,V_b,V_a] :
      ( V_a = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),V_q)),V_r)
     => ( ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b))
           => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_r))
              & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_r),V_b)) ) )
          & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b))
           => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b),V_r))
              & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_r),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) ) ) )
       => hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,V_b),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_q),V_r))) ) ) )).

fof(fact_zmult1__lemma,axiom,(
    ! [V_a,V_r,V_q,V_c,V_b] :
      ( hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_b,V_c),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_q),V_r)))
     => ( V_c != c_Groups_Ozero__class_Ozero(tc_Int_Oint)
       => hBOOL(hAPP(c_Divides_Odivmod__int__rel(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),V_b),V_c),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),V_q)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),V_r)),V_c))),c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),V_r),V_c)))) ) ) )).

fof(fact_posDivAlg_Opinduct,axiom,(
    ! [V_P_2,V_a1_2,V_a0_2] :
      ( hBOOL(hAPP(c_Wellfounded_Oaccp(tc_prod(tc_Int_Oint,tc_Int_Oint),c_Divides_OposDivAlg__rel),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_a0_2),V_a1_2)))
     => ( ! [B_a,B_b] :
            ( hBOOL(hAPP(c_Wellfounded_Oaccp(tc_prod(tc_Int_Oint,tc_Int_Oint),c_Divides_OposDivAlg__rel),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),B_a),B_b)))
           => ( ( ~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B_a),B_b))
                    | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B_b),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) )
               => hBOOL(hAPP(hAPP(V_P_2,B_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),B_b))) )
             => hBOOL(hAPP(hAPP(V_P_2,B_a),B_b)) ) )
       => hBOOL(hAPP(hAPP(V_P_2,V_a0_2),V_a1_2)) ) ) )).

fof(fact_nat__gcd_Opsimps,axiom,(
    ! [V_y_2,V_x_2] :
      ( hBOOL(hAPP(c_Wellfounded_Oaccp(tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Nitpick_Onat__gcd__rel),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_x_2),V_y_2)))
     => ( ( V_y_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
         => c_Nitpick_Onat__gcd(V_x_2,V_y_2) = V_x_2 )
        & ( V_y_2 != c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
         => c_Nitpick_Onat__gcd(V_x_2,V_y_2) = c_Nitpick_Onat__gcd(V_y_2,c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_x_2,V_y_2)) ) ) ) )).

fof(fact_aset_I4_J,axiom,(
    ! [V_A_2,V_ta_2,V_D_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_D_2))
     => ( hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),V_ta_2),V_A_2))
       => ! [B_x] :
            ( ! [B_xa] :
                ( hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xa),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),V_D_2)))
               => ! [B_xb] :
                    ( hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xb),V_A_2))
                   => B_x != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B_xb),B_xa) ) )
           => ( B_x != V_ta_2
             => hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B_x),V_D_2) != V_ta_2 ) ) ) ) )).

fof(fact_bset_I4_J,axiom,(
    ! [V_B_2,V_ta_2,V_D_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_D_2))
     => ( hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),V_ta_2),V_B_2))
       => ! [B_x] :
            ( ! [B_xa] :
                ( hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xa),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),V_D_2)))
               => ! [B_xb] :
                    ( hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xb),V_B_2))
                   => B_x != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B_xb),B_xa) ) )
           => ( B_x != V_ta_2
             => hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B_x),V_D_2) != V_ta_2 ) ) ) ) )).

fof(fact_bset_I7_J,axiom,(
    ! [V_B_2,V_ta_2,V_D_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_D_2))
     => ( hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),V_ta_2),V_B_2))
       => ! [B_x] :
            ( ! [B_xa] :
                ( hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xa),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),V_D_2)))
               => ! [B_xb] :
                    ( hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xb),V_B_2))
                   => B_x != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B_xb),B_xa) ) )
           => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_ta_2),B_x))
             => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_ta_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B_x),V_D_2))) ) ) ) ) )).

fof(fact_bset_I3_J,axiom,(
    ! [V_B_2,V_ta_2,V_D_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_D_2))
     => ( hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_ta_2),c_Groups_Oone__class_Oone(tc_Int_Oint))),V_B_2))
       => ( ! [B_x] :
              ( hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_x),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),V_D_2)))
             => ! [B_xa] :
                  ( hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xa),V_B_2))
                 => V_ta_2 != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B_xa),B_x) ) )
         => hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_ta_2),V_D_2) = V_ta_2 ) ) ) )).

fof(fact_aset_I3_J,axiom,(
    ! [V_A_2,V_ta_2,V_D_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_D_2))
     => ( hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_ta_2),c_Groups_Oone__class_Oone(tc_Int_Oint))),V_A_2))
       => ( ! [B_x] :
              ( hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_x),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),V_D_2)))
             => ! [B_xa] :
                  ( hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xa),V_A_2))
                 => V_ta_2 != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B_xa),B_x) ) )
         => hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_ta_2),V_D_2) = V_ta_2 ) ) ) )).

fof(fact_aset_I5_J,axiom,(
    ! [V_A_2,V_ta_2,V_D_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_D_2))
     => ( hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),V_ta_2),V_A_2))
       => ! [B_x] :
            ( ! [B_xa] :
                ( hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xa),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),V_D_2)))
               => ! [B_xb] :
                    ( hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xb),V_A_2))
                   => B_x != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B_xb),B_xa) ) )
           => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B_x),V_ta_2))
             => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B_x),V_D_2)),V_ta_2)) ) ) ) ) )).

fof(fact_in__measure,axiom,(
    ! [V_f_2,V_y_2,V_x_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_y_2)),hAPP(c_Wellfounded_Omeasure(T_b),V_f_2)))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(V_f_2,V_x_2)),hAPP(V_f_2,V_y_2))) ) )).

fof(fact_bset_I6_J,axiom,(
    ! [V_ta_2,V_B_2,V_D_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_D_2))
     => ! [B_x] :
          ( ! [B_xa] :
              ( hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xa),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),V_D_2)))
             => ! [B_xb] :
                  ( hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xb),V_B_2))
                 => B_x != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B_xb),B_xa) ) )
         => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B_x),V_ta_2))
           => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B_x),V_D_2)),V_ta_2)) ) ) ) )).

fof(fact_aset_I8_J,axiom,(
    ! [V_ta_2,V_A_2,V_D_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_D_2))
     => ! [B_x] :
          ( ! [B_xa] :
              ( hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xa),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),V_D_2)))
             => ! [B_xb] :
                  ( hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xb),V_A_2))
                 => B_x != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B_xb),B_xa) ) )
         => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_ta_2),B_x))
           => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_ta_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B_x),V_D_2))) ) ) ) )).

fof(fact_nat__gcd_Opinduct,axiom,(
    ! [V_P_2,V_a1_2,V_a0_2] :
      ( hBOOL(hAPP(c_Wellfounded_Oaccp(tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Nitpick_Onat__gcd__rel),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_a0_2),V_a1_2)))
     => ( ! [B_x,B_y] :
            ( hBOOL(hAPP(c_Wellfounded_Oaccp(tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Nitpick_Onat__gcd__rel),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),B_x),B_y)))
           => ( ( B_y != c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
               => hBOOL(hAPP(hAPP(V_P_2,B_y),c_Divides_Odiv__class_Omod(tc_Nat_Onat,B_x,B_y))) )
             => hBOOL(hAPP(hAPP(V_P_2,B_x),B_y)) ) )
       => hBOOL(hAPP(hAPP(V_P_2,V_a0_2),V_a1_2)) ) ) )).

fof(fact_mlex__leq,axiom,(
    ! [V_R_2,T_b,V_y_2,V_x_2,V_f_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(V_f_2,V_x_2)),hAPP(V_f_2,V_y_2)))
     => ( hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_y_2)),V_R_2))
       => hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_y_2)),c_Wellfounded_Omlex__prod(T_b,V_f_2,V_R_2))) ) ) )).

fof(fact_mlex__less,axiom,(
    ! [V_R_2,T_b,V_y_2,V_x_2,V_f_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(V_f_2,V_x_2)),hAPP(V_f_2,V_y_2)))
     => hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_y_2)),c_Wellfounded_Omlex__prod(T_b,V_f_2,V_R_2))) ) )).

fof(fact_periodic__finite__ex,axiom,(
    ! [V_P_2,V_d_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_d_2))
     => ( ! [B_x,B_k] :
            ( hBOOL(hAPP(V_P_2,B_x))
          <=> hBOOL(hAPP(V_P_2,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B_k),V_d_2)))) )
       => ( ? [B_x1] : hBOOL(hAPP(V_P_2,B_x1))
        <=> ? [B_x] :
              ( hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_x),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),V_d_2)))
              & hBOOL(hAPP(V_P_2,B_x)) ) ) ) ) )).

fof(fact_aset_I7_J,axiom,(
    ! [V_ta_2,V_A_2,V_D_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_D_2))
     => ! [B_x] :
          ( ! [B_xa] :
              ( hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xa),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),V_D_2)))
             => ! [B_xb] :
                  ( hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xb),V_A_2))
                 => B_x != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B_xb),B_xa) ) )
         => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_ta_2),B_x))
           => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_ta_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B_x),V_D_2))) ) ) ) )).

fof(fact_bset_I5_J,axiom,(
    ! [V_ta_2,V_B_2,V_D_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_D_2))
     => ! [B_x] :
          ( ! [B_xa] :
              ( hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xa),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),V_D_2)))
             => ! [B_xb] :
                  ( hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xb),V_B_2))
                 => B_x != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B_xb),B_xa) ) )
         => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B_x),V_ta_2))
           => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B_x),V_D_2)),V_ta_2)) ) ) ) )).

fof(fact_divmod__int__def,axiom,(
    ! [V_b,V_a] :
      ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a))
       => ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b))
           => c_Divides_Odivmod__int(V_a,V_b) = c_Divides_OposDivAlg(V_a,V_b) )
          & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b))
           => ( ( V_a = c_Groups_Ozero__class_Ozero(tc_Int_Oint)
               => c_Divides_Odivmod__int(V_a,V_b) = hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Groups_Ozero__class_Ozero(tc_Int_Oint)) )
              & ( V_a != c_Groups_Ozero__class_Ozero(tc_Int_Oint)
               => c_Divides_Odivmod__int(V_a,V_b) = hAPP(c_Divides_OnegateSnd,c_Divides_OnegDivAlg(hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_b))) ) ) ) ) )
      & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a))
       => ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b))
           => c_Divides_Odivmod__int(V_a,V_b) = c_Divides_OnegDivAlg(V_a,V_b) )
          & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b))
           => c_Divides_Odivmod__int(V_a,V_b) = hAPP(c_Divides_OnegateSnd,c_Divides_OposDivAlg(hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_b))) ) ) ) ) )).

fof(fact_mod__pos__pos__1__number__of,axiom,(
    ! [V_w] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)))
     => c_Divides_Odiv__class_Omod(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)) = hAPP(c_Product__Type_Osnd(tc_Int_Oint,tc_Int_Oint),c_Divides_OposDivAlg(c_Groups_Oone__class_Oone(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w))) ) )).

fof(fact_in__lex__prod,axiom,(
    ! [V_s_2,V_r_2,V_b_H_2,V_a_H_2,V_b_2,V_a_2,T_c,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(tc_prod(tc_prod(T_b,T_c),tc_prod(T_b,T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_prod(T_b,T_c),tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_a_2),V_b_2)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_a_H_2),V_b_H_2))),c_Wellfounded_Olex__prod(T_b,T_c,V_r_2,V_s_2)))
    <=> ( hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_a_H_2)),V_r_2))
        | ( V_a_2 = V_a_H_2
          & hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_b_2),V_b_H_2)),V_s_2)) ) ) ) )).

fof(fact_mod__int__def,axiom,(
    ! [V_b,V_a] : c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b) = hAPP(c_Product__Type_Osnd(tc_Int_Oint,tc_Int_Oint),c_Divides_Odivmod__int(V_a,V_b)) )).

fof(fact_mod__pos__neg,axiom,(
    ! [V_b,V_a] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
       => c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b) = hAPP(c_Product__Type_Osnd(tc_Int_Oint,tc_Int_Oint),hAPP(c_Divides_OnegateSnd,c_Divides_OnegDivAlg(hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_b)))) ) ) )).

fof(fact_negateSnd__eq,axiom,(
    ! [V_r,V_q] : hAPP(c_Divides_OnegateSnd,hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_q),V_r)) = hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_q),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_r)) )).

fof(fact_mod__neg__neg,axiom,(
    ! [V_b,V_a] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_a),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_b),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
       => c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b) = hAPP(c_Product__Type_Osnd(tc_Int_Oint,tc_Int_Oint),hAPP(c_Divides_OnegateSnd,c_Divides_OposDivAlg(hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_b)))) ) ) )).

fof(fact_divmod__int__rel__neg,axiom,(
    ! [V_qr,V_b,V_a] :
      ( hBOOL(hAPP(c_Divides_Odivmod__int__rel(hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_b)),V_qr))
     => hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,V_b),hAPP(c_Divides_OnegateSnd,V_qr))) ) )).

fof(fact_mod__neg__pos,axiom,(
    ! [V_b,V_a] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_a),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b))
       => c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b) = hAPP(c_Product__Type_Osnd(tc_Int_Oint,tc_Int_Oint),c_Divides_OnegDivAlg(V_a,V_b)) ) ) )).

fof(fact_divmod__int__correct,axiom,(
    ! [V_a,V_b] :
      ( V_b != c_Groups_Ozero__class_Ozero(tc_Int_Oint)
     => hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,V_b),c_Divides_Odivmod__int(V_a,V_b))) ) )).

fof(fact_mod__pos__neg__1__number__of,axiom,(
    ! [V_w] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
     => c_Divides_Odiv__class_Omod(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)) = hAPP(c_Product__Type_Osnd(tc_Int_Oint,tc_Int_Oint),hAPP(c_Divides_OnegateSnd,c_Divides_OnegDivAlg(hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w))))) ) )).

fof(fact_mod__pos__pos,axiom,(
    ! [V_b,V_a] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b))
       => c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b) = hAPP(c_Product__Type_Osnd(tc_Int_Oint,tc_Int_Oint),c_Divides_OposDivAlg(V_a,V_b)) ) ) )).

fof(fact_divmod__int__mod__div,axiom,(
    ! [V_q,V_p] : c_Divides_Odivmod__int(V_p,V_q) = hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_p),V_q)),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_p,V_q)) )).

fof(fact_same__fstI,axiom,(
    ! [T_b,V_R_2,V_y_2,V_y_H_2,T_c,V_x_2,V_P_2] :
      ( hBOOL(hAPP(V_P_2,V_x_2))
     => ( hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_y_H_2),V_y_2)),hAPP(V_R_2,V_x_2)))
       => hBOOL(hAPP(hAPP(c_member(tc_prod(tc_prod(T_b,T_c),tc_prod(T_b,T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_prod(T_b,T_c),tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_x_2),V_y_H_2)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_x_2),V_y_2))),c_Recdef_Osame__fst(T_b,T_c,V_P_2,V_R_2))) ) ) )).

fof(fact_div__pos__neg__1__number__of,axiom,(
    ! [V_w] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
     => hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)) = hAPP(c_Product__Type_Ofst(tc_Int_Oint,tc_Int_Oint),hAPP(c_Divides_OnegateSnd,c_Divides_OnegDivAlg(hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w))))) ) )).

fof(fact_div__neg__neg,axiom,(
    ! [V_b,V_a] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_a),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_b),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
       => hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b) = hAPP(c_Product__Type_Ofst(tc_Int_Oint,tc_Int_Oint),hAPP(c_Divides_OnegateSnd,c_Divides_OposDivAlg(hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_b)))) ) ) )).

fof(fact_snd__eqD,axiom,(
    ! [V_a,V_y,V_x,T_a,T_b] :
      ( hAPP(c_Product__Type_Osnd(T_b,T_a),hAPP(hAPP(c_Product__Type_OPair(T_b,T_a),V_x),V_y)) = V_a
     => V_y = V_a ) )).

fof(fact_snd__conv,axiom,(
    ! [V_b,V_a,T_a,T_b] : hAPP(c_Product__Type_Osnd(T_b,T_a),hAPP(hAPP(c_Product__Type_OPair(T_b,T_a),V_a),V_b)) = V_b )).

fof(fact_Pair__fst__snd__eq,axiom,(
    ! [T_c,T_b,V_ta_2,V_s_2] :
      ( V_s_2 = V_ta_2
    <=> ( hAPP(c_Product__Type_Ofst(T_b,T_c),V_s_2) = hAPP(c_Product__Type_Ofst(T_b,T_c),V_ta_2)
        & hAPP(c_Product__Type_Osnd(T_b,T_c),V_s_2) = hAPP(c_Product__Type_Osnd(T_b,T_c),V_ta_2) ) ) )).

fof(fact_prod__eqI,axiom,(
    ! [V_q,V_p,T_b,T_a] :
      ( hAPP(c_Product__Type_Ofst(T_a,T_b),V_p) = hAPP(c_Product__Type_Ofst(T_a,T_b),V_q)
     => ( hAPP(c_Product__Type_Osnd(T_a,T_b),V_p) = hAPP(c_Product__Type_Osnd(T_a,T_b),V_q)
       => V_p = V_q ) ) )).

fof(fact_fst__eqD,axiom,(
    ! [V_a,V_y,V_x,T_b,T_a] :
      ( hAPP(c_Product__Type_Ofst(T_a,T_b),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_x),V_y)) = V_a
     => V_x = V_a ) )).

fof(fact_fst__conv,axiom,(
    ! [V_b,V_a,T_b,T_a] : hAPP(c_Product__Type_Ofst(T_a,T_b),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_a),V_b)) = V_a )).

fof(fact_surjective__pairing,axiom,(
    ! [T_b,T_a,V_t] : V_t = hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),hAPP(c_Product__Type_Ofst(T_a,T_b),V_t)),hAPP(c_Product__Type_Osnd(T_a,T_b),V_t)) )).

fof(fact_pair__collapse,axiom,(
    ! [V_p,T_b,T_a] : hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),hAPP(c_Product__Type_Ofst(T_a,T_b),V_p)),hAPP(c_Product__Type_Osnd(T_a,T_b),V_p)) = V_p )).

fof(fact_div__int__def,axiom,(
    ! [V_b,V_a] : hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b) = hAPP(c_Product__Type_Ofst(tc_Int_Oint,tc_Int_Oint),c_Divides_Odivmod__int(V_a,V_b)) )).

fof(fact_mod__nat__def,axiom,(
    ! [V_n,V_m] : c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n) = hAPP(c_Product__Type_Osnd(tc_Nat_Onat,tc_Nat_Onat),c_Divides_Odivmod__nat(V_m,V_n)) )).

fof(fact_Pair__inject,axiom,(
    ! [V_b_H,V_a_H,V_b,V_a,T_b,T_a] :
      ( hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_a),V_b) = hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_a_H),V_b_H)
     => ~ ( V_a = V_a_H
         => V_b != V_b_H ) ) )).

fof(fact_Pair__eq,axiom,(
    ! [V_b_H_2,V_a_H_2,V_b_2,V_a_2,T_c,T_b] :
      ( hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_a_2),V_b_2) = hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_a_H_2),V_b_H_2)
    <=> ( V_a_2 = V_a_H_2
        & V_b_2 = V_b_H_2 ) ) )).

fof(fact_split__paired__All,axiom,(
    ! [T_c,T_b,V_P_2] :
      ( ! [B_x1] : hBOOL(hAPP(V_P_2,B_x1))
    <=> ! [B_a,B_b] : hBOOL(hAPP(V_P_2,hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),B_a),B_b))) ) )).

fof(fact_div__neg__pos,axiom,(
    ! [V_b,V_a] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_a),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b))
       => hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b) = hAPP(c_Product__Type_Ofst(tc_Int_Oint,tc_Int_Oint),c_Divides_OnegDivAlg(V_a,V_b)) ) ) )).

fof(fact_div__pos__pos,axiom,(
    ! [V_b,V_a] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b))
       => hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b) = hAPP(c_Product__Type_Ofst(tc_Int_Oint,tc_Int_Oint),c_Divides_OposDivAlg(V_a,V_b)) ) ) )).

fof(fact_div__pos__pos__1__number__of,axiom,(
    ! [V_w] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)))
     => hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)) = hAPP(c_Product__Type_Ofst(tc_Int_Oint,tc_Int_Oint),c_Divides_OposDivAlg(c_Groups_Oone__class_Oone(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w))) ) )).

fof(fact_div__pos__neg,axiom,(
    ! [V_b,V_a] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))
       => hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b) = hAPP(c_Product__Type_Ofst(tc_Int_Oint,tc_Int_Oint),hAPP(c_Divides_OnegateSnd,c_Divides_OnegDivAlg(hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_b)))) ) ) )).

fof(fact_prod__size__simp,axiom,(
    ! [V_p_2,V_g_2,V_f_2,T_c,T_b] : c_Product__Type_Oprod_Oprod__size(T_b,T_c,V_f_2,V_g_2,V_p_2) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(V_f_2,hAPP(c_Product__Type_Ofst(T_b,T_c),V_p_2))),hAPP(V_g_2,hAPP(c_Product__Type_Osnd(T_b,T_c),V_p_2)))),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) )).

fof(fact_conjI__realizer,axiom,(
    ! [T_c,T_b,V_q_2,V_Q_2,V_p_2,V_P_2] :
      ( hBOOL(hAPP(V_P_2,V_p_2))
     => ( hBOOL(hAPP(V_Q_2,V_q_2))
       => ( hBOOL(hAPP(V_P_2,hAPP(c_Product__Type_Ofst(T_b,T_c),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_p_2),V_q_2))))
          & hBOOL(hAPP(V_Q_2,hAPP(c_Product__Type_Osnd(T_b,T_c),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_p_2),V_q_2)))) ) ) ) )).

fof(fact_exI__realizer,axiom,(
    ! [T_b,T_c,V_x_2,V_y_2,V_P_2] :
      ( hBOOL(hAPP(hAPP(V_P_2,V_y_2),V_x_2))
     => hBOOL(hAPP(hAPP(V_P_2,hAPP(c_Product__Type_Osnd(T_c,T_b),hAPP(hAPP(c_Product__Type_OPair(T_c,T_b),V_x_2),V_y_2))),hAPP(c_Product__Type_Ofst(T_c,T_b),hAPP(hAPP(c_Product__Type_OPair(T_c,T_b),V_x_2),V_y_2)))) ) )).

fof(fact_prod_Orecs,axiom,(
    ! [V_b_2,V_a_2,V_f1_2,T_b,T_d,T_c] : c_Product__Type_Oprod_Oprod__rec(T_c,T_d,T_b,V_f1_2,hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_a_2),V_b_2)) = hAPP(hAPP(V_f1_2,V_a_2),V_b_2) )).

fof(fact_in__finite__psubset,axiom,(
    ! [V_B_2,V_A_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),c_Wellfounded_Ofinite__psubset(T_b)))
    <=> ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))
        & hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) ) ) )).

fof(fact_div__nat__def,axiom,(
    ! [V_n,V_m] : hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),V_n) = hAPP(c_Product__Type_Ofst(tc_Nat_Onat,tc_Nat_Onat),c_Divides_Odivmod__nat(V_m,V_n)) )).

fof(fact_divmod__nat__rel__def,axiom,(
    ! [V_qr_2,V_n_2,V_m_2] :
      ( hBOOL(hAPP(c_Divides_Odivmod__nat__rel(V_m_2,V_n_2),V_qr_2))
    <=> ( V_m_2 = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Product__Type_Ofst(tc_Nat_Onat,tc_Nat_Onat),V_qr_2)),V_n_2)),hAPP(c_Product__Type_Osnd(tc_Nat_Onat,tc_Nat_Onat),V_qr_2))
        & ( V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
         => hAPP(c_Product__Type_Ofst(tc_Nat_Onat,tc_Nat_Onat),V_qr_2) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) )
        & ( V_n_2 != c_Groups_Ozero__class_Ozero(tc_Nat_Onat)
         => ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n_2))
             => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Product__Type_Osnd(tc_Nat_Onat,tc_Nat_Onat),V_qr_2)))
                & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Product__Type_Osnd(tc_Nat_Onat,tc_Nat_Onat),V_qr_2)),V_n_2)) ) )
            & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n_2))
             => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),hAPP(c_Product__Type_Osnd(tc_Nat_Onat,tc_Nat_Onat),V_qr_2)))
                & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Product__Type_Osnd(tc_Nat_Onat,tc_Nat_Onat),V_qr_2)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) ) ) ) ) ) ) )).

fof(fact_range,axiom,(
    ! [V_s,V_k] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),V_k))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(c_Product__Type_Ofst(tc_Code__Numeral_Ocode__numeral,tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral)),hAPP(c_Random_Orange(V_k),V_s))),V_k)) ) )).

fof(fact_SetInterval_Otransfer__nat__int__set__functions_I1_J,axiom,(
    ! [V_n_2] : hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),V_n_2) = hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n_2))) )).

fof(fact_pair__lessI2,axiom,(
    ! [V_ta_2,V_s_2,V_b_2,V_a_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_a_2),V_b_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_s_2),V_ta_2))
       => hBOOL(hAPP(hAPP(c_member(tc_prod(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat))),hAPP(hAPP(c_Product__Type_OPair(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_a_2),V_s_2)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_b_2),V_ta_2))),c_FunDef_Opair__less)) ) ) )).

fof(fact_Nitpick_Ozero__frac__def,axiom,(
    ! [T_a] : c_Nitpick_Ozero__frac(T_a) = c_Nitpick_OAbs__Frac(T_a,hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Groups_Oone__class_Oone(tc_Int_Oint))) )).

fof(fact_finite__atMost,axiom,(
    ! [V_k_2] : hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),V_k_2))) )).

fof(fact_atMost__eq__iff,axiom,(
    ! [V_y_2,V_x_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hAPP(c_SetInterval_Oord__class_OatMost(T_b),V_x_2) = hAPP(c_SetInterval_Oord__class_OatMost(T_b),V_y_2)
      <=> V_x_2 = V_y_2 ) ) )).

fof(fact_atLeast0AtMost,axiom,(
    ! [V_n_2] : c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2) = hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),V_n_2) )).

fof(fact_card__atMost,axiom,(
    ! [V_u_2] : hAPP(c_Finite__Set_Ocard(tc_Nat_Onat),hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),V_u_2)) = hAPP(c_Nat_OSuc,V_u_2) )).

fof(fact_atMost__Suc,axiom,(
    ! [V_k_2] : hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_k_2)) = hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_k_2)),hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),V_k_2)) )).

fof(fact_atMost__iff,axiom,(
    ! [V_k_2,V_i_2,T_b] :
      ( class_Orderings_Oord(T_b)
     => ( hBOOL(hAPP(hAPP(c_member(T_b),V_i_2),hAPP(c_SetInterval_Oord__class_OatMost(T_b),V_k_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_i_2),V_k_2)) ) ) )).

fof(fact_atMost__subset__iff,axiom,(
    ! [V_y_2,V_x_2,T_b] :
      ( class_Orderings_Oorder(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatMost(T_b),V_x_2)),hAPP(c_SetInterval_Oord__class_OatMost(T_b),V_y_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),V_y_2)) ) ) )).

fof(fact_atMost__0,axiom,(
    hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool))) )).

fof(fact_pair__lessI1,axiom,(
    ! [V_ta_2,V_s_2,V_b_2,V_a_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_a_2),V_b_2))
     => hBOOL(hAPP(hAPP(c_member(tc_prod(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat))),hAPP(hAPP(c_Product__Type_OPair(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_a_2),V_s_2)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_b_2),V_ta_2))),c_FunDef_Opair__less)) ) )).

fof(fact_Nitpick_Oone__frac__def,axiom,(
    ! [T_a] : c_Nitpick_Oone__frac(T_a) = c_Nitpick_OAbs__Frac(T_a,hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),c_Groups_Oone__class_Oone(tc_Int_Oint))) )).

fof(fact_Nitpick_Onumber__of__frac__def,axiom,(
    ! [V_n,T_a] : c_Nitpick_Onumber__of__frac(T_a,V_n) = c_Nitpick_OAbs__Frac(T_a,hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_n),c_Groups_Oone__class_Oone(tc_Int_Oint))) )).

fof(fact_pair__leqI2,axiom,(
    ! [V_ta_2,V_s_2,V_b_2,V_a_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_a_2),V_b_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_s_2),V_ta_2))
       => hBOOL(hAPP(hAPP(c_member(tc_prod(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat))),hAPP(hAPP(c_Product__Type_OPair(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_a_2),V_s_2)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_b_2),V_ta_2))),c_FunDef_Opair__leq)) ) ) )).

fof(fact_smax__insertI,axiom,(
    ! [V_X_2,V_x_2,V_Y_2,V_y_2] :
      ( hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),V_y_2),V_Y_2))
     => ( hBOOL(hAPP(hAPP(c_member(tc_prod(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat))),hAPP(hAPP(c_Product__Type_OPair(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat)),V_x_2),V_y_2)),c_FunDef_Opair__less))
       => ( hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),V_X_2),V_Y_2)),c_FunDef_Omax__strict))
         => hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),V_x_2),V_X_2)),V_Y_2)),c_FunDef_Omax__strict)) ) ) ) )).

fof(fact_smax__emptyI,axiom,(
    ! [V_Y_2] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),V_Y_2))
     => ( V_Y_2 != c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))
       => hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))),V_Y_2)),c_FunDef_Omax__strict)) ) ) )).

fof(fact_pair__leqI1,axiom,(
    ! [V_ta_2,V_s_2,V_b_2,V_a_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_a_2),V_b_2))
     => hBOOL(hAPP(hAPP(c_member(tc_prod(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat))),hAPP(hAPP(c_Product__Type_OPair(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_a_2),V_s_2)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_b_2),V_ta_2))),c_FunDef_Opair__leq)) ) )).

fof(fact_wmin__insertI,axiom,(
    ! [V_YS_2,V_y_2,V_XS_2,V_x_2] :
      ( hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),V_x_2),V_XS_2))
     => ( hBOOL(hAPP(hAPP(c_member(tc_prod(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat))),hAPP(hAPP(c_Product__Type_OPair(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat)),V_x_2),V_y_2)),c_FunDef_Opair__leq))
       => ( hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),V_XS_2),V_YS_2)),c_FunDef_Omin__weak))
         => hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),V_XS_2),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),V_y_2),V_YS_2))),c_FunDef_Omin__weak)) ) ) ) )).

fof(fact_wmax__insertI,axiom,(
    ! [V_XS_2,V_x_2,V_YS_2,V_y_2] :
      ( hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),V_y_2),V_YS_2))
     => ( hBOOL(hAPP(hAPP(c_member(tc_prod(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat))),hAPP(hAPP(c_Product__Type_OPair(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat)),V_x_2),V_y_2)),c_FunDef_Opair__leq))
       => ( hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),V_XS_2),V_YS_2)),c_FunDef_Omax__weak))
         => hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),V_x_2),V_XS_2)),V_YS_2)),c_FunDef_Omax__weak)) ) ) ) )).

fof(fact_smin__insertI,axiom,(
    ! [V_YS_2,V_y_2,V_XS_2,V_x_2] :
      ( hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),V_x_2),V_XS_2))
     => ( hBOOL(hAPP(hAPP(c_member(tc_prod(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat))),hAPP(hAPP(c_Product__Type_OPair(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat)),V_x_2),V_y_2)),c_FunDef_Opair__less))
       => ( hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),V_XS_2),V_YS_2)),c_FunDef_Omin__strict))
         => hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),V_XS_2),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),V_y_2),V_YS_2))),c_FunDef_Omin__strict)) ) ) ) )).

fof(fact_smin__emptyI,axiom,(
    ! [V_X_2] :
      ( V_X_2 != c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))
     => hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),V_X_2),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)))),c_FunDef_Omin__strict)) ) )).

fof(fact_less__by__empty,axiom,(
    ! [V_B_2,T_b,V_A_2] :
      ( V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_A_2),V_B_2)) ) )).

fof(fact_wmax__emptyI,axiom,(
    ! [V_X_2] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),V_X_2))
     => hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))),V_X_2)),c_FunDef_Omax__weak)) ) )).

fof(fact_wmin__emptyI,axiom,(
    ! [V_X_2] : hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),V_X_2),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)))),c_FunDef_Omin__weak)) )).

fof(fact_min__rpair__set,axiom,(
    c_FunDef_Oreduction__pair(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),hAPP(hAPP(c_Product__Type_OPair(tc_fun(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),tc_HOL_Obool),tc_fun(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),tc_HOL_Obool)),c_FunDef_Omin__strict),c_FunDef_Omin__weak)) )).

fof(fact_max__rpair__set,axiom,(
    c_FunDef_Oreduction__pair(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),hAPP(hAPP(c_Product__Type_OPair(tc_fun(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),tc_HOL_Obool),tc_fun(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),tc_HOL_Obool)),c_FunDef_Omax__strict),c_FunDef_Omax__weak)) )).

fof(fact_min__strict__def,axiom,(
    c_FunDef_Omin__strict = c_Wellfounded_Omin__ext(tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_FunDef_Opair__less) )).

fof(fact_less__than__iff,axiom,(
    ! [V_y_2,V_x_2] :
      ( hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_x_2),V_y_2)),c_Wellfounded_Oless__than))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_x_2),V_y_2)) ) )).

fof(fact_pair__less__def,axiom,(
    c_FunDef_Opair__less = c_Wellfounded_Olex__prod(tc_Nat_Onat,tc_Nat_Onat,c_Wellfounded_Oless__than,c_Wellfounded_Oless__than) )).

fof(fact_min__weak__def,axiom,(
    c_FunDef_Omin__weak = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),tc_HOL_Obool)),c_Wellfounded_Omin__ext(tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_FunDef_Opair__leq)),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)))),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),tc_HOL_Obool)))) )).

fof(fact_intrel__iff,axiom,(
    ! [V_v_2,V_u_2,V_y_2,V_x_2] :
      ( hBOOL(hAPP(hAPP(c_member(tc_prod(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat))),hAPP(hAPP(c_Product__Type_OPair(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_x_2),V_y_2)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_u_2),V_v_2))),c_Int_Ointrel))
    <=> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_x_2),V_v_2) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_u_2),V_y_2) ) )).

fof(fact_rp__inv__image__rp,axiom,(
    ! [V_f_2,T_c,V_P_2,T_b] :
      ( c_FunDef_Oreduction__pair(T_b,V_P_2)
     => c_FunDef_Oreduction__pair(T_c,hAPP(hAPP(c_FunDef_Orp__inv__image(T_b,T_c),V_P_2),V_f_2)) ) )).

fof(fact_measure__def,axiom,(
    ! [T_b] : c_Wellfounded_Omeasure(T_b) = hAPP(c_Relation_Oinv__image(tc_Nat_Onat,T_b),c_Wellfounded_Oless__than) )).

fof(fact_sup1E,axiom,(
    ! [V_x_2,V_B_2,V_A_2,T_b] :
      ( hBOOL(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2),V_x_2))
     => ( ~ hBOOL(hAPP(V_A_2,V_x_2))
       => hBOOL(hAPP(V_B_2,V_x_2)) ) ) )).

fof(fact_sup1CI,axiom,(
    ! [T_b,V_A_2,V_x_2,V_B_2] :
      ( ( ~ hBOOL(hAPP(V_B_2,V_x_2))
       => hBOOL(hAPP(V_A_2,V_x_2)) )
     => hBOOL(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2),V_x_2)) ) )).

fof(fact_UnE,axiom,(
    ! [V_B_2,V_A_2,V_c_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)))
     => ( ~ hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_A_2))
       => hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_B_2)) ) ) )).

fof(fact_UnCI,axiom,(
    ! [V_A_2,V_B_2,V_c_2,T_b] :
      ( ( ~ hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_B_2))
       => hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_A_2)) )
     => hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) ) )).

fof(fact_sup__eq__bot__iff,axiom,(
    ! [V_y_2,V_x_2,T_b] :
      ( class_Lattices_Obounded__lattice__bot(T_b)
     => ( hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_b),V_x_2),V_y_2) = c_Orderings_Obot__class_Obot(T_b)
      <=> ( V_x_2 = c_Orderings_Obot__class_Obot(T_b)
          & V_y_2 = c_Orderings_Obot__class_Obot(T_b) ) ) ) )).

fof(fact_sup__bot__right,axiom,(
    ! [V_x,T_a] :
      ( class_Lattices_Obounded__lattice__bot(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),c_Orderings_Obot__class_Obot(T_a)) = V_x ) )).

fof(fact_sup__bot__left,axiom,(
    ! [V_x,T_a] :
      ( class_Lattices_Obounded__lattice__bot(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),c_Orderings_Obot__class_Obot(T_a)),V_x) = V_x ) )).

fof(fact_Un__empty,axiom,(
    ! [V_B_2,V_A_2,T_b] :
      ( hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
    <=> ( V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
        & V_B_2 = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) ) ) )).

fof(fact_Un__empty__right,axiom,(
    ! [V_A_2,T_b] : hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) = V_A_2 )).

fof(fact_Un__empty__left,axiom,(
    ! [V_B_2,T_b] : hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))),V_B_2) = V_B_2 )).

fof(fact_le__supE,axiom,(
    ! [V_x,V_b,V_a,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),V_b)),V_x))
       => ~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_x))
           => ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),V_x)) ) ) ) )).

fof(fact_sup__mono,axiom,(
    ! [V_d,V_b,V_c,V_a,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_c))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),V_d))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),V_b)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_c),V_d))) ) ) ) )).

fof(fact_sup__least,axiom,(
    ! [V_z,V_x,V_y,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),V_x))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_z),V_x))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_y),V_z)),V_x)) ) ) ) )).

fof(fact_le__supI,axiom,(
    ! [V_b,V_x,V_a,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_x))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),V_x))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),V_b)),V_x)) ) ) ) )).

fof(fact_sup__absorb1,axiom,(
    ! [V_x,V_y,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),V_x))
       => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y) = V_x ) ) )).

fof(fact_sup__absorb2,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y))
       => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y) = V_y ) ) )).

fof(fact_le__supI2,axiom,(
    ! [V_a,V_b,V_x,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_b))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),V_b))) ) ) )).

fof(fact_le__supI1,axiom,(
    ! [V_b,V_a,V_x,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_a))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),V_b))) ) ) )).

fof(fact_le__sup__iff,axiom,(
    ! [V_z_2,V_y_2,V_x_2,T_b] :
      ( class_Lattices_Osemilattice__sup(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_b),V_x_2),V_y_2)),V_z_2))
      <=> ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),V_z_2))
          & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_y_2),V_z_2)) ) ) ) )).

fof(fact_le__iff__sup,axiom,(
    ! [V_y_2,V_x_2,T_b] :
      ( class_Lattices_Osemilattice__sup(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),V_y_2))
      <=> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_b),V_x_2),V_y_2) = V_y_2 ) ) )).

fof(fact_sup__ge2,axiom,(
    ! [V_x,V_y,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y))) ) )).

fof(fact_inf__sup__ord_I4_J,axiom,(
    ! [V_x,V_y,T_a] :
      ( class_Lattices_Olattice(T_a)
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y))) ) )).

fof(fact_sup__ge1,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y))) ) )).

fof(fact_inf__sup__ord_I3_J,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Lattices_Olattice(T_a)
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y))) ) )).

fof(fact_Un__upper1,axiom,(
    ! [V_B_2,V_A_2,T_b] : hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) )).

fof(fact_Un__upper2,axiom,(
    ! [V_A_2,V_B_2,T_b] : hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) )).

fof(fact_subset__Un__eq,axiom,(
    ! [V_B_2,V_A_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))
    <=> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) = V_B_2 ) )).

fof(fact_Un__absorb1,axiom,(
    ! [V_B_2,V_A_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))
     => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) = V_B_2 ) )).

fof(fact_Un__absorb2,axiom,(
    ! [V_A_2,V_B_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2))
     => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) = V_A_2 ) )).

fof(fact_Un__least,axiom,(
    ! [V_B_2,V_C_2,V_A_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_C_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),V_C_2)) ) ) )).

fof(fact_Un__mono,axiom,(
    ! [V_D_2,V_B_2,V_C_2,V_A_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_C_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_D_2))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_D_2))) ) ) )).

fof(fact_Un__Pow__subset,axiom,(
    ! [V_B_2,V_A_2,T_b] : hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)),hAPP(c_Set_OPow(T_b),V_A_2)),hAPP(c_Set_OPow(T_b),V_B_2))),hAPP(c_Set_OPow(T_b),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)))) )).

fof(fact_Un__iff,axiom,(
    ! [V_B_2,V_A_2,V_c_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)))
    <=> ( hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_A_2))
        | hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_B_2)) ) ) )).

fof(fact_UnI1,axiom,(
    ! [V_B_2,V_A_2,V_c_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_A_2))
     => hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) ) )).

fof(fact_UnI2,axiom,(
    ! [V_A_2,V_B_2,V_c_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_B_2))
     => hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) ) )).

fof(fact_Un__insert__left,axiom,(
    ! [V_C_2,V_B_2,V_a_2,T_b] : hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_B_2)),V_C_2) = hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2)) )).

fof(fact_Un__insert__right,axiom,(
    ! [V_B_2,V_a_2,V_A_2,T_b] : hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_B_2)) = hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) )).

fof(fact_image__Un,axiom,(
    ! [V_B_2,V_A_2,V_f_2,T_b,T_c] : hAPP(c_Set_Oimage(T_c,T_b,V_f_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_c,T_b,V_f_2),V_A_2)),hAPP(c_Set_Oimage(T_c,T_b,V_f_2),V_B_2)) )).

fof(fact_less__supI2,axiom,(
    ! [V_a,V_b,V_x,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_b))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),V_b))) ) ) )).

fof(fact_less__supI1,axiom,(
    ! [V_b,V_a,V_x,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_a))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),V_b))) ) ) )).

fof(fact_Un__Diff,axiom,(
    ! [V_C_2,V_B_2,V_A_2,T_b] : hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),V_C_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_C_2)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2)) )).

fof(fact_Un__Diff__cancel2,axiom,(
    ! [V_A_2,V_B_2,T_b] : hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2)),V_A_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2) )).

fof(fact_Un__Diff__cancel,axiom,(
    ! [V_B_2,V_A_2,T_b] : hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) )).

fof(fact_Sup__fin_Oidem,axiom,(
    ! [V_x,T_a] :
      ( class_Lattices_Olattice(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_x) = V_x ) )).

fof(fact_sup1I1,axiom,(
    ! [V_B_2,T_b,V_x_2,V_A_2] :
      ( hBOOL(hAPP(V_A_2,V_x_2))
     => hBOOL(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2),V_x_2)) ) )).

fof(fact_sup1I2,axiom,(
    ! [V_A_2,T_b,V_x_2,V_B_2] :
      ( hBOOL(hAPP(V_B_2,V_x_2))
     => hBOOL(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2),V_x_2)) ) )).

fof(fact_Un__absorb,axiom,(
    ! [V_A_2,T_b] : hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_A_2) = V_A_2 )).

fof(fact_Un__commute,axiom,(
    ! [V_B_2,V_A_2,T_b] : hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2) )).

fof(fact_Un__left__absorb,axiom,(
    ! [V_B_2,V_A_2,T_b] : hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) )).

fof(fact_Un__left__commute,axiom,(
    ! [V_C_2,V_B_2,V_A_2,T_b] : hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_B_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_C_2)) )).

fof(fact_Un__assoc,axiom,(
    ! [V_C_2,V_B_2,V_A_2,T_b] : hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),V_C_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2)) )).

fof(fact_bex__Un,axiom,(
    ! [V_P_2,V_B_2,V_A_2,T_b] :
      ( ? [B_x] :
          ( hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)))
          & hBOOL(hAPP(V_P_2,B_x)) )
    <=> ( ? [B_x] :
            ( hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2))
            & hBOOL(hAPP(V_P_2,B_x)) )
        | ? [B_x] :
            ( hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_B_2))
            & hBOOL(hAPP(V_P_2,B_x)) ) ) ) )).

fof(fact_ball__Un,axiom,(
    ! [V_P_2,V_B_2,V_A_2,T_b] :
      ( ! [B_x] :
          ( hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)))
         => hBOOL(hAPP(V_P_2,B_x)) )
    <=> ( ! [B_x] :
            ( hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2))
           => hBOOL(hAPP(V_P_2,B_x)) )
        & ! [B_x] :
            ( hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_B_2))
           => hBOOL(hAPP(V_P_2,B_x)) ) ) ) )).

fof(fact_sup_Oidem,axiom,(
    ! [V_a,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),V_a) = V_a ) )).

fof(fact_sup__idem,axiom,(
    ! [V_x,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_x) = V_x ) )).

fof(fact_sup_Ocommute,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),V_b) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_b),V_a) ) )).

fof(fact_inf__sup__aci_I5_J,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Lattices_Olattice(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_y),V_x) ) )).

fof(fact_sup__commute,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_y),V_x) ) )).

fof(fact_sup_Oleft__idem,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),V_b)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),V_b) ) )).

fof(fact_inf__sup__aci_I8_J,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Lattices_Olattice(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y) ) )).

fof(fact_sup__left__idem,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y) ) )).

fof(fact_sup_Oleft__commute,axiom,(
    ! [V_c,V_a,V_b,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_b),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),V_c)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_b),V_c)) ) )).

fof(fact_inf__sup__aci_I7_J,axiom,(
    ! [V_z,V_y,V_x,T_a] :
      ( class_Lattices_Olattice(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_y),V_z)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_y),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_z)) ) )).

fof(fact_sup__left__commute,axiom,(
    ! [V_z,V_y,V_x,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_y),V_z)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_y),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_z)) ) )).

fof(fact_sup_Oassoc,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),V_b)),V_c) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_b),V_c)) ) )).

fof(fact_inf__sup__aci_I6_J,axiom,(
    ! [V_z,V_y,V_x,T_a] :
      ( class_Lattices_Olattice(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y)),V_z) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_y),V_z)) ) )).

fof(fact_sup__assoc,axiom,(
    ! [V_z,V_y,V_x,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y)),V_z) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_y),V_z)) ) )).

fof(fact_sup__apply,axiom,(
    ! [V_x_2,V_g_2,V_f_2,T_c,T_b] :
      ( class_Lattices_Olattice(T_b)
     => hAPP(hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,T_b)),V_f_2),V_g_2),V_x_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_b),hAPP(V_f_2,V_x_2)),hAPP(V_g_2,V_x_2)) ) )).

fof(fact_sup__max,axiom,(
    ! [T_b] :
      ( ( class_Lattices_Osemilattice__sup(T_b)
        & class_Orderings_Olinorder(T_b) )
     => c_Lattices_Osemilattice__sup__class_Osup(T_b) = c_Orderings_Oord__class_Omax(T_b) ) )).

fof(fact_finite__Un,axiom,(
    ! [V_Ga_2,V_F_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_F_2),V_Ga_2)))
    <=> ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_F_2))
        & hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_Ga_2)) ) ) )).

fof(fact_finite__UnI,axiom,(
    ! [V_Ga_2,V_F_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_F_2))
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_Ga_2))
       => hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_F_2),V_Ga_2))) ) ) )).

fof(fact_fun__left__comm__idem__sup,axiom,(
    ! [T_b] :
      ( class_Lattices_Osemilattice__sup(T_b)
     => c_Finite__Set_Ofun__left__comm__idem(T_b,T_b,c_Lattices_Osemilattice__sup__class_Osup(T_b)) ) )).

fof(fact_insert__is__Un,axiom,(
    ! [V_A_2,V_a_2,T_b] : hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))),V_A_2) )).

fof(fact_union__fold__insert,axiom,(
    ! [V_B_2,V_A_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
     => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) = hAPP(hAPP(c_Finite__Set_Ofold(T_b,tc_fun(T_b,tc_HOL_Obool),c_Set_Oinsert(T_b)),V_B_2),V_A_2) ) )).

fof(fact_Diff__subset__conv,axiom,(
    ! [V_C_2,V_B_2,V_A_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),V_C_2))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2))) ) )).

fof(fact_Diff__partition,axiom,(
    ! [V_B_2,V_A_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))
     => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2)) = V_B_2 ) )).

fof(fact_Compl__Diff__eq,axiom,(
    ! [V_B_2,V_A_2,T_b] : hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),V_A_2)),V_B_2) )).

fof(fact_folding__image__simple__idem_Ounion__idem,axiom,(
    ! [V_B_2,V_A_2,V_F_2,V_g_2,V_z_2,V_f_2,T_c,T_b] :
      ( c_Finite__Set_Ofolding__image__simple__idem(T_b,T_c,V_f_2,V_z_2,V_g_2,V_F_2)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2))
       => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2))
         => hAPP(V_F_2,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(V_f_2,hAPP(V_F_2,V_A_2)),hAPP(V_F_2,V_B_2)) ) ) ) )).

fof(fact_fold__sup__insert,axiom,(
    ! [V_a_2,V_b_2,V_A_2,T_b] :
      ( class_Lattices_Osemilattice__sup(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Lattices_Osemilattice__sup__class_Osup(T_b)),V_b_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_b),V_a_2),hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Lattices_Osemilattice__sup__class_Osup(T_b)),V_b_2),V_A_2)) ) ) )).

fof(fact_folding__one__idem_Ounion__idem,axiom,(
    ! [V_B_2,V_A_2,V_F_2,V_f_2,T_b] :
      ( c_Finite__Set_Ofolding__one__idem(T_b,V_f_2,V_F_2)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
         => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2))
           => ( V_B_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
             => hAPP(V_F_2,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(V_f_2,hAPP(V_F_2,V_A_2)),hAPP(V_F_2,V_B_2)) ) ) ) ) ) )).

fof(fact_sup__le__fold__sup,axiom,(
    ! [V_b_2,V_a_2,V_A_2,T_b] :
      ( class_Lattices_Osemilattice__sup(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_b),V_a_2),V_b_2)),hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Lattices_Osemilattice__sup__class_Osup(T_b)),V_b_2),V_A_2))) ) ) ) )).

fof(fact_fold1__Un2,axiom,(
    ! [V_B_2,V_A_2,T_b] :
      ( class_Lattices_Oab__semigroup__idem__mult(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
         => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2))
           => ( V_B_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
             => hAPP(c_Finite__Set_Ofold1(T_b,c_Groups_Otimes__class_Otimes(T_b)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(c_Finite__Set_Ofold1(T_b,c_Groups_Otimes__class_Otimes(T_b)),V_A_2)),hAPP(c_Finite__Set_Ofold1(T_b,c_Groups_Otimes__class_Otimes(T_b)),V_B_2)) ) ) ) ) ) )).

fof(fact_Min__Un,axiom,(
    ! [V_B_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
         => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2))
           => ( V_B_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
             => c_Big__Operators_Olinorder__class_OMin(T_b,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),c_Big__Operators_Olinorder__class_OMin(T_b,V_A_2)),c_Big__Operators_Olinorder__class_OMin(T_b,V_B_2)) ) ) ) ) ) )).

fof(fact_Max__Un,axiom,(
    ! [V_B_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
         => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2))
           => ( V_B_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
             => c_Big__Operators_Olinorder__class_OMax(T_b,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),c_Big__Operators_Olinorder__class_OMax(T_b,V_A_2)),c_Big__Operators_Olinorder__class_OMax(T_b,V_B_2)) ) ) ) ) ) )).

fof(fact_min__max_OInf__fin_Ounion__idem,axiom,(
    ! [V_B_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
         => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2))
           => ( V_B_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
             => c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),V_A_2)),c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),V_B_2)) ) ) ) ) ) )).

fof(fact_min__max_OSup__fin_Ounion__idem,axiom,(
    ! [V_B_2,V_A_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
         => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2))
           => ( V_B_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
             => c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),V_A_2)),c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),V_B_2)) ) ) ) ) ) )).

fof(fact_max__weak__def,axiom,(
    c_FunDef_Omax__weak = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),tc_HOL_Obool)),c_Wellfounded_Omax__ext(tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_FunDef_Opair__leq)),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)))),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),tc_HOL_Obool)))) )).

fof(fact_in__inv__image,axiom,(
    ! [V_f_2,V_r_2,T_c,V_y_2,V_x_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_y_2)),hAPP(hAPP(c_Relation_Oinv__image(T_c,T_b),V_r_2),V_f_2)))
    <=> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),hAPP(V_f_2,V_x_2)),hAPP(V_f_2,V_y_2))),V_r_2)) ) )).

fof(fact_ivl__disj__un_I19_J,axiom,(
    ! [V_u_2,V_m_2,V_l_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_l_2),V_m_2))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_m_2),V_u_2))
         => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanLessThan(T_b,V_l_2,V_m_2)),c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_m_2,V_u_2)) = c_SetInterval_Oord__class_OgreaterThanAtMost(T_b,V_l_2,V_u_2) ) ) ) )).

fof(fact_ivl__disj__un_I4_J,axiom,(
    ! [V_u_2,V_l_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_l_2),V_u_2))
       => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanLessThan(T_b,V_l_2,V_u_2)),hAPP(hAPP(c_Set_Oinsert(T_b),V_u_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) = c_SetInterval_Oord__class_OgreaterThanAtMost(T_b,V_l_2,V_u_2) ) ) )).

fof(fact_finite__greaterThanAtMost,axiom,(
    ! [V_u_2,V_l_2] : hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Nat_Onat,V_l_2,V_u_2))) )).

fof(fact_finite__greaterThanAtMost__int,axiom,(
    ! [V_u_2,V_l_2] : hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Int_Oint,V_l_2,V_u_2))) )).

fof(fact_Nat__Transfer_Otransfer__int__nat__set__function__closures_I2_J,axiom,(
    ! [V_B_2,V_A_2] :
      ( c_Nat__Transfer_Onat__set(V_A_2)
     => ( c_Nat__Transfer_Onat__set(V_B_2)
       => c_Nat__Transfer_Onat__set(hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Int_Oint,tc_HOL_Obool)),V_A_2),V_B_2)) ) ) )).

fof(fact_sup__nat__def,axiom,(
    c_Lattices_Osemilattice__sup__class_Osup(tc_Nat_Onat) = c_Orderings_Oord__class_Omax(tc_Nat_Onat) )).

fof(fact_Nat__Transfer_Otransfer__nat__int__set__functions_I3_J,axiom,(
    ! [V_B_2,V_A_2] : hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),V_A_2),V_B_2) = hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),V_A_2)),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),V_B_2))) )).

fof(fact_Nat__Transfer_Otransfer__int__nat__set__functions_I3_J,axiom,(
    ! [V_B_2,V_A_2] :
      ( c_Nat__Transfer_Onat__set(V_A_2)
     => ( c_Nat__Transfer_Onat__set(V_B_2)
       => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Int_Oint,tc_HOL_Obool)),V_A_2),V_B_2) = hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),V_A_2)),hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),V_B_2))) ) ) )).

fof(fact_greaterThanAtMost__empty,axiom,(
    ! [V_k_2,V_l_2,T_b] :
      ( class_Orderings_Oorder(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_l_2),V_k_2))
       => c_SetInterval_Oord__class_OgreaterThanAtMost(T_b,V_k_2,V_l_2) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) ) ) )).

fof(fact_greaterThanAtMost__empty__iff,axiom,(
    ! [V_l_2,V_k_2,T_b] :
      ( class_Orderings_Oorder(T_b)
     => ( c_SetInterval_Oord__class_OgreaterThanAtMost(T_b,V_k_2,V_l_2) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
      <=> ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_k_2),V_l_2)) ) ) )).

fof(fact_greaterThanAtMost__empty__iff2,axiom,(
    ! [V_l_2,V_k_2,T_b] :
      ( class_Orderings_Oorder(T_b)
     => ( c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = c_SetInterval_Oord__class_OgreaterThanAtMost(T_b,V_k_2,V_l_2)
      <=> ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_k_2),V_l_2)) ) ) )).

fof(fact_ivl__disj__un_I20_J,axiom,(
    ! [V_u_2,V_m_2,V_l_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_l_2),V_m_2))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_m_2),V_u_2))
         => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanAtMost(T_b,V_l_2,V_m_2)),c_SetInterval_Oord__class_OgreaterThanAtMost(T_b,V_m_2,V_u_2)) = c_SetInterval_Oord__class_OgreaterThanAtMost(T_b,V_l_2,V_u_2) ) ) ) )).

fof(fact_atLeastSucAtMost__greaterThanAtMost,axiom,(
    ! [V_u_2,V_l_2] : c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_l_2),V_u_2) = c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Nat_Onat,V_l_2,V_u_2) )).

fof(fact_card__greaterThanAtMost,axiom,(
    ! [V_u_2,V_l_2] : hAPP(c_Finite__Set_Ocard(tc_Nat_Onat),c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Nat_Onat,V_l_2,V_u_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_u_2),V_l_2) )).

fof(fact_Pow__insert,axiom,(
    ! [V_A_2,V_a_2,T_b] : hAPP(c_Set_OPow(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)),hAPP(c_Set_OPow(T_b),V_A_2)),hAPP(c_Set_Oimage(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),hAPP(c_Set_Oinsert(T_b),V_a_2)),hAPP(c_Set_OPow(T_b),V_A_2))) )).

fof(fact_ivl__disj__un_I22_J,axiom,(
    ! [V_u_2,V_m_2,V_l_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_l_2),V_m_2))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_m_2),V_u_2))
         => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_l_2,V_m_2)),c_SetInterval_Oord__class_OgreaterThanAtMost(T_b,V_m_2,V_u_2)) = c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_l_2,V_u_2) ) ) ) )).

fof(fact_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,(
    ! [V_u_2,V_l_2] : c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_l_2),c_Groups_Oone__class_Oone(tc_Int_Oint)),V_u_2) = c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Int_Oint,V_l_2,V_u_2) )).

fof(fact_ivl__disj__un_I9_J,axiom,(
    ! [V_u_2,V_l_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_l_2),V_u_2))
       => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatMost(T_b),V_l_2)),c_SetInterval_Oord__class_OgreaterThanAtMost(T_b,V_l_2,V_u_2)) = hAPP(c_SetInterval_Oord__class_OatMost(T_b),V_u_2) ) ) )).

fof(fact_card__greaterThanAtMost__int,axiom,(
    ! [V_u_2,V_l_2] : hAPP(c_Finite__Set_Ocard(tc_Int_Oint),c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Int_Oint,V_l_2,V_u_2)) = hAPP(c_Int_Onat,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_u_2),V_l_2)) )).

fof(fact_max__strict__def,axiom,(
    c_FunDef_Omax__strict = c_Wellfounded_Omax__ext(tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_FunDef_Opair__less) )).

fof(fact_ivl__disj__un_I16_J,axiom,(
    ! [V_u_2,V_m_2,V_l_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_l_2),V_m_2))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_m_2),V_u_2))
         => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanAtMost(T_b,V_l_2,V_m_2)),c_SetInterval_Oord__class_OgreaterThanLessThan(T_b,V_m_2,V_u_2)) = c_SetInterval_Oord__class_OgreaterThanLessThan(T_b,V_l_2,V_u_2) ) ) ) )).

fof(fact_max__ext__additive,axiom,(
    ! [V_D_2,V_C_2,V_R_2,V_B_2,V_A_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),c_Wellfounded_Omax__ext(T_b,V_R_2)))
     => ( hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_D_2)),c_Wellfounded_Omax__ext(T_b,V_R_2)))
       => hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_C_2)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_D_2))),c_Wellfounded_Omax__ext(T_b,V_R_2))) ) ) )).

fof(fact_ivl__disj__un_I5_J,axiom,(
    ! [V_u_2,V_l_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_l_2),V_u_2))
       => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_b),V_l_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))),c_SetInterval_Oord__class_OgreaterThanAtMost(T_b,V_l_2,V_u_2)) = c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_l_2,V_u_2) ) ) )).

fof(fact_irrefl__def,axiom,(
    ! [V_r_2,T_b] :
      ( c_Relation_Oirrefl(T_b,V_r_2)
    <=> ! [B_x] : ~ hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),B_x),B_x)),V_r_2)) ) )).

fof(fact_Field__insert,axiom,(
    ! [V_r_2,V_b_2,V_a_2,T_b] : hAPP(c_Relation_OField(T_b),hAPP(hAPP(c_Set_Oinsert(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_b_2)),V_r_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_b_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))),hAPP(c_Relation_OField(T_b),V_r_2)) )).

fof(fact_Sup__fin_Oremove,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( class_Lattices_Olattice(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
         => ( ( hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
             => c_Big__Operators_Olattice__class_OSup__fin(T_b,V_A_2) = V_x_2 )
            & ( hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
             => c_Big__Operators_Olattice__class_OSup__fin(T_b,V_A_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_b),V_x_2),c_Big__Operators_Olattice__class_OSup__fin(T_b,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))) ) ) ) ) ) )).

fof(fact_ivl__disj__un_I7_J,axiom,(
    ! [V_u_2,V_l_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_l_2),V_u_2))
       => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatMost(T_b),V_l_2)),c_SetInterval_Oord__class_OgreaterThanLessThan(T_b,V_l_2,V_u_2)) = hAPP(c_SetInterval_Oord__class_OlessThan(T_b),V_u_2) ) ) )).

fof(fact_finite__lessThan,axiom,(
    ! [V_k_2] : hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),V_k_2))) )).

fof(fact_lessThan__eq__iff,axiom,(
    ! [V_y_2,V_x_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hAPP(c_SetInterval_Oord__class_OlessThan(T_b),V_x_2) = hAPP(c_SetInterval_Oord__class_OlessThan(T_b),V_y_2)
      <=> V_x_2 = V_y_2 ) ) )).

fof(fact_card__lessThan,axiom,(
    ! [V_u_2] : hAPP(c_Finite__Set_Ocard(tc_Nat_Onat),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),V_u_2)) = V_u_2 )).

fof(fact_Sup__fin_Osingleton,axiom,(
    ! [V_x_2,T_b] :
      ( class_Lattices_Olattice(T_b)
     => c_Big__Operators_Olattice__class_OSup__fin(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) = V_x_2 ) )).

fof(fact_lessThan__0,axiom,(
    hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) )).

fof(fact_lessThan__Suc__atMost,axiom,(
    ! [V_k_2] : hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_k_2)) = hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),V_k_2) )).

fof(fact_lessThan__Suc,axiom,(
    ! [V_k_2] : hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_k_2)) = hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),V_k_2),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),V_k_2)) )).

fof(fact_lessThan__iff,axiom,(
    ! [V_k_2,V_i_2,T_b] :
      ( class_Orderings_Oord(T_b)
     => ( hBOOL(hAPP(hAPP(c_member(T_b),V_i_2),hAPP(c_SetInterval_Oord__class_OlessThan(T_b),V_k_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_i_2),V_k_2)) ) ) )).

fof(fact_lessThan__subset__iff,axiom,(
    ! [V_y_2,V_x_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(T_b),V_x_2)),hAPP(c_SetInterval_Oord__class_OlessThan(T_b),V_y_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),V_y_2)) ) ) )).

fof(fact_lessThan__strict__subset__iff,axiom,(
    ! [V_n_2,V_m_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(T_b),V_m_2)),hAPP(c_SetInterval_Oord__class_OlessThan(T_b),V_n_2)))
      <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_m_2),V_n_2)) ) ) )).

fof(fact_Field__empty,axiom,(
    ! [T_b] : hAPP(c_Relation_OField(T_b),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) )).

fof(fact_mono__Field,axiom,(
    ! [V_s_2,V_r_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_r_2),V_s_2))
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Relation_OField(T_b),V_r_2)),hAPP(c_Relation_OField(T_b),V_s_2))) ) )).

fof(fact_Field__Un,axiom,(
    ! [V_s_2,V_r_2,T_b] : hAPP(c_Relation_OField(T_b),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_r_2),V_s_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Relation_OField(T_b),V_r_2)),hAPP(c_Relation_OField(T_b),V_s_2)) )).

fof(fact_Sup__fin_Oin__idem,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( class_Lattices_Olattice(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
         => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_b),V_x_2),c_Big__Operators_Olattice__class_OSup__fin(T_b,V_A_2)) = c_Big__Operators_Olattice__class_OSup__fin(T_b,V_A_2) ) ) ) )).

fof(fact_finite__Field,axiom,(
    ! [V_r_2,T_b] :
      ( hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(T_b,T_b)),V_r_2))
     => hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),hAPP(c_Relation_OField(T_b),V_r_2))) ) )).

fof(fact_Sup__fin_OF__eq,axiom,(
    ! [V_A_2,T_b] :
      ( class_Lattices_Olattice(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => c_Big__Operators_Olattice__class_OSup__fin(T_b,V_A_2) = hAPP(c_Finite__Set_Ofold1(T_b,c_Lattices_Osemilattice__sup__class_Osup(T_b)),V_A_2) ) ) )).

fof(fact_single__Diff__lessThan,axiom,(
    ! [V_k_2,T_b] :
      ( class_Orderings_Oorder(T_b)
     => hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_b),V_k_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))),hAPP(c_SetInterval_Oord__class_OlessThan(T_b),V_k_2)) = hAPP(hAPP(c_Set_Oinsert(T_b),V_k_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) ) )).

fof(fact_Sup__fin_Oinsert__idem,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( class_Lattices_Olattice(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
         => c_Big__Operators_Olattice__class_OSup__fin(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_b),V_x_2),c_Big__Operators_Olattice__class_OSup__fin(T_b,V_A_2)) ) ) ) )).

fof(fact_Sup__fin_Osubset__idem,axiom,(
    ! [V_B_2,V_A_2,T_b] :
      ( class_Lattices_Olattice(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( V_B_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
         => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2))
           => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_b),c_Big__Operators_Olattice__class_OSup__fin(T_b,V_B_2)),c_Big__Operators_Olattice__class_OSup__fin(T_b,V_A_2)) = c_Big__Operators_Olattice__class_OSup__fin(T_b,V_A_2) ) ) ) ) )).

fof(fact_Sup__fin_Ounion__idem,axiom,(
    ! [V_B_2,V_A_2,T_b] :
      ( class_Lattices_Olattice(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
         => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2))
           => ( V_B_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
             => c_Big__Operators_Olattice__class_OSup__fin(T_b,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_b),c_Big__Operators_Olattice__class_OSup__fin(T_b,V_A_2)),c_Big__Operators_Olattice__class_OSup__fin(T_b,V_B_2)) ) ) ) ) ) )).

fof(fact_Sup__fin_Oeq__fold__idem_H,axiom,(
    ! [V_a_2,V_A_2,T_b] :
      ( class_Lattices_Olattice(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => c_Big__Operators_Olattice__class_OSup__fin(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2)) = hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Lattices_Osemilattice__sup__class_Osup(T_b)),V_a_2),V_A_2) ) ) )).

fof(fact_lessThan__Suc__eq__insert__0,axiom,(
    ! [V_n_2] : hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_n_2)) = hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Nat_Onat,c_Nat_OSuc),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),V_n_2))) )).

fof(fact_Sup__fin_Oinsert,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( class_Lattices_Olattice(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( ~ hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
         => ( V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
           => c_Big__Operators_Olattice__class_OSup__fin(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_b),V_x_2),c_Big__Operators_Olattice__class_OSup__fin(T_b,V_A_2)) ) ) ) ) )).

fof(fact_Sup__fin_Oinsert__remove,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( class_Lattices_Olattice(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( ( hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
           => c_Big__Operators_Olattice__class_OSup__fin(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = V_x_2 )
          & ( hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
           => c_Big__Operators_Olattice__class_OSup__fin(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_b),V_x_2),c_Big__Operators_Olattice__class_OSup__fin(T_b,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))) ) ) ) ) )).

fof(fact_Sup__fin_Oeq__fold_H,axiom,(
    ! [V_x_2,V_A_2,T_b] :
      ( class_Lattices_Olattice(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( ~ hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))
         => c_Big__Operators_Olattice__class_OSup__fin(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Lattices_Osemilattice__sup__class_Osup(T_b)),V_x_2),V_A_2) ) ) ) )).

fof(fact_ivl__disj__un_I2_J,axiom,(
    ! [V_u_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(T_b),V_u_2)),hAPP(hAPP(c_Set_Oinsert(T_b),V_u_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) = hAPP(c_SetInterval_Oord__class_OatMost(T_b),V_u_2) ) )).

fof(fact_ivl__disj__un_I10_J,axiom,(
    ! [V_u_2,V_l_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_l_2),V_u_2))
       => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(T_b),V_l_2)),c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_l_2,V_u_2)) = hAPP(c_SetInterval_Oord__class_OatMost(T_b),V_u_2) ) ) )).

fof(fact_Sup__fin_Oclosed,axiom,(
    ! [V_A_2,T_b] :
      ( class_Lattices_Olattice(T_b)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
         => ( ! [B_x,B_y] : hBOOL(hAPP(hAPP(c_member(T_b),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_b),B_x),B_y)),hAPP(hAPP(c_Set_Oinsert(T_b),B_x),hAPP(hAPP(c_Set_Oinsert(T_b),B_y),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))))
           => hBOOL(hAPP(hAPP(c_member(T_b),c_Big__Operators_Olattice__class_OSup__fin(T_b,V_A_2)),V_A_2)) ) ) ) ) )).

fof(fact_Sup__fin_Ohom__commute,axiom,(
    ! [V_N_2,V_h_2,T_b] :
      ( class_Lattices_Olattice(T_b)
     => ( ! [B_x,B_y] : hAPP(V_h_2,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_b),B_x),B_y)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_b),hAPP(V_h_2,B_x)),hAPP(V_h_2,B_y))
       => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_N_2))
         => ( V_N_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
           => hAPP(V_h_2,c_Big__Operators_Olattice__class_OSup__fin(T_b,V_N_2)) = c_Big__Operators_Olattice__class_OSup__fin(T_b,hAPP(c_Set_Oimage(T_b,T_b,V_h_2),V_N_2)) ) ) ) ) )).

fof(fact_ivl__disj__un_I18_J,axiom,(
    ! [V_u_2,V_m_2,V_l_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_l_2),V_m_2))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_m_2),V_u_2))
         => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_l_2,V_m_2)),c_SetInterval_Oord__class_OgreaterThanLessThan(T_b,V_m_2,V_u_2)) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_l_2),V_u_2) ) ) ) )).

fof(fact_equiv__intrel__iff,axiom,(
    ! [V_y_2,V_x_2] :
      ( hAPP(c_Relation_OImage(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Int_Ointrel),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)))) = hAPP(c_Relation_OImage(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Int_Ointrel),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),V_y_2),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))))
    <=> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat))),hAPP(hAPP(c_Product__Type_OPair(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat)),V_x_2),V_y_2)),c_Int_Ointrel)) ) )).

fof(fact_finite__atLeastLessThan,axiom,(
    ! [V_u_2,V_l_2] : hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_l_2),V_u_2))) )).

fof(fact_finite__atLeastLessThan__int,axiom,(
    ! [V_u_2,V_l_2] : hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,V_l_2),V_u_2))) )).

fof(fact_Image__Un,axiom,(
    ! [V_B_2,V_A_2,V_R_2,T_b,T_c] : hAPP(c_Relation_OImage(T_c,T_b,V_R_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_c,T_b,V_R_2),V_A_2)),hAPP(c_Relation_OImage(T_c,T_b,V_R_2),V_B_2)) )).

fof(fact_atLeast0LessThan,axiom,(
    ! [V_n_2] : hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n_2) = hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),V_n_2) )).

fof(fact_atLeastLessThan__inj_I2_J,axiom,(
    ! [V_d_2,V_c_2,V_b_2,V_a_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_a_2),V_b_2) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_c_2),V_d_2)
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),V_b_2))
         => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),V_d_2))
           => V_b_2 = V_d_2 ) ) ) ) )).

fof(fact_atLeastLessThan__inj_I1_J,axiom,(
    ! [V_d_2,V_c_2,V_b_2,V_a_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_a_2),V_b_2) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_c_2),V_d_2)
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),V_b_2))
         => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),V_d_2))
           => V_a_2 = V_c_2 ) ) ) ) )).

fof(fact_atLeastLessThan__eq__iff,axiom,(
    ! [V_d_2,V_c_2,V_b_2,V_a_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),V_b_2))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),V_d_2))
         => ( hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_a_2),V_b_2) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_c_2),V_d_2)
          <=> ( V_a_2 = V_c_2
              & V_b_2 = V_d_2 ) ) ) ) ) )).

fof(fact_Image__empty,axiom,(
    ! [V_R_2,T_b,T_c] : hAPP(c_Relation_OImage(T_c,T_b,V_R_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) )).

fof(fact_ex__nat__less__eq,axiom,(
    ! [V_P_2,V_n_2] :
      ( ? [B_m] :
          ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
          & hBOOL(hAPP(V_P_2,B_m)) )
    <=> ? [B_x] :
          ( hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n_2)))
          & hBOOL(hAPP(V_P_2,B_x)) ) ) )).

fof(fact_all__nat__less__eq,axiom,(
    ! [V_P_2,V_n_2] :
      ( ! [B_m] :
          ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2))
         => hBOOL(hAPP(V_P_2,B_m)) )
    <=> ! [B_x] :
          ( hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n_2)))
         => hBOOL(hAPP(V_P_2,B_x)) ) ) )).

fof(fact_atLeastLessThanSuc__atLeastAtMost,axiom,(
    ! [V_u_2,V_l_2] : hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_l_2),hAPP(c_Nat_OSuc,V_u_2)) = c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_l_2,V_u_2) )).

fof(fact_atLeastLessThan0,axiom,(
    ! [V_m_2] : hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) )).

fof(fact_card__atLeastLessThan,axiom,(
    ! [V_u_2,V_l_2] : hAPP(c_Finite__Set_Ocard(tc_Nat_Onat),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_l_2),V_u_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_u_2),V_l_2) )).

fof(fact_Image__mono,axiom,(
    ! [V_A_2,V_A_H_2,V_r_2,V_r_H_2,T_c,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)),V_r_H_2),V_r_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_H_2),V_A_2))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_b,T_c,V_r_H_2),V_A_H_2)),hAPP(c_Relation_OImage(T_b,T_c,V_r_2),V_A_2))) ) ) )).

fof(fact_Un__Image,axiom,(
    ! [V_A_2,V_S_2,V_R_2,T_b,T_c] : hAPP(c_Relation_OImage(T_c,T_b,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_c,T_b),tc_HOL_Obool)),V_R_2),V_S_2)),V_A_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_c,T_b,V_R_2),V_A_2)),hAPP(c_Relation_OImage(T_c,T_b,V_S_2),V_A_2)) )).

fof(fact_image__Suc__atLeastLessThan,axiom,(
    ! [V_j_2,V_i_2] : hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Nat_Onat,c_Nat_OSuc),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_i_2),V_j_2)) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_i_2)),hAPP(c_Nat_OSuc,V_j_2)) )).

fof(fact_atLeastLessThan__empty,axiom,(
    ! [V_a_2,V_b_2,T_b] :
      ( class_Orderings_Oorder(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_b_2),V_a_2))
       => hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_a_2),V_b_2) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) ) ) )).

fof(fact_atLeastLessThan__empty__iff,axiom,(
    ! [V_b_2,V_a_2,T_b] :
      ( class_Orderings_Oorder(T_b)
     => ( hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_a_2),V_b_2) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
      <=> ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),V_b_2)) ) ) )).

fof(fact_atLeastLessThan__empty__iff2,axiom,(
    ! [V_b_2,V_a_2,T_b] :
      ( class_Orderings_Oorder(T_b)
     => ( c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_a_2),V_b_2)
      <=> ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),V_b_2)) ) ) )).

fof(fact_atLeastLessThan__subset__iff,axiom,(
    ! [V_d_2,V_c_2,V_b_2,V_a_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_a_2),V_b_2)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_c_2),V_d_2)))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_b_2),V_a_2))
          | ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_c_2),V_a_2))
            & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_b_2),V_d_2)) ) ) ) ) )).

fof(fact_ivl__disj__un_I17_J,axiom,(
    ! [V_u_2,V_m_2,V_l_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_l_2),V_m_2))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_m_2),V_u_2))
         => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_l_2),V_m_2)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_m_2),V_u_2)) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_l_2),V_u_2) ) ) ) )).

fof(fact_ivl__diff,axiom,(
    ! [V_m_2,V_n_2,V_i_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_i_2),V_n_2))
       => hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_i_2),V_m_2)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_i_2),V_n_2)) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_n_2),V_m_2) ) ) )).

fof(fact_atLeastSucLessThan__greaterThanLessThan,axiom,(
    ! [V_u_2,V_l_2] : hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_l_2)),V_u_2) = c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Nat_Onat,V_l_2,V_u_2) )).

fof(fact_finite__atLeastZeroLessThan__int,axiom,(
    ! [V_u_2] : hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_u_2))) )).

fof(fact_setsum__add__nat__ivl,axiom,(
    ! [V_f_2,V_p_2,V_n_2,V_m_2,T_b] :
      ( class_Groups_Ocomm__monoid__add(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n_2),V_p_2))
         => hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2),V_n_2))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_n_2),V_p_2))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2),V_p_2)) ) ) ) )).

fof(fact_setsum__diff__nat__ivl,axiom,(
    ! [V_f_2,V_p_2,V_n_2,V_m_2,T_b] :
      ( class_Groups_Oab__group__add(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n_2),V_p_2))
         => hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2),V_p_2))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2),V_n_2))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_n_2),V_p_2)) ) ) ) )).

fof(fact_rev__ImageI,axiom,(
    ! [V_r_2,V_b_2,T_c,V_A_2,V_a_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2))
     => ( hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_a_2),V_b_2)),V_r_2))
       => hBOOL(hAPP(hAPP(c_member(T_c),V_b_2),hAPP(c_Relation_OImage(T_b,T_c,V_r_2),V_A_2))) ) ) )).

fof(fact_Image__iff,axiom,(
    ! [V_A_2,V_r_2,T_c,V_b_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),hAPP(c_Relation_OImage(T_c,T_b,V_r_2),V_A_2)))
    <=> ? [B_x] :
          ( hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2))
          & hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_b),B_x),V_b_2)),V_r_2)) ) ) )).

fof(fact_atLeastLessThan__singleton,axiom,(
    ! [V_m_2] : hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2),hAPP(c_Nat_OSuc,V_m_2)) = hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),V_m_2),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool))) )).

fof(fact_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,(
    ! [V_u_2,V_l_2] : hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,V_l_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_u_2),c_Groups_Oone__class_Oone(tc_Int_Oint))) = c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,V_l_2,V_u_2) )).

fof(fact_ivl__disj__un_I21_J,axiom,(
    ! [V_u_2,V_m_2,V_l_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_l_2),V_m_2))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_m_2),V_u_2))
         => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_l_2),V_m_2)),c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_m_2,V_u_2)) = c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_l_2,V_u_2) ) ) ) )).

fof(fact_atLeastLessThan__add__Un,axiom,(
    ! [V_k_2,V_j_2,V_i_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i_2),V_j_2))
     => hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_i_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_j_2),V_k_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_i_2),V_j_2)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_j_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_j_2),V_k_2))) ) )).

fof(fact_ivl__disj__un_I8_J,axiom,(
    ! [V_u_2,V_l_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_l_2),V_u_2))
       => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(T_b),V_l_2)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_l_2),V_u_2)) = hAPP(c_SetInterval_Oord__class_OlessThan(T_b),V_u_2) ) ) )).

fof(fact_subset__card__intvl__is__intvl,axiom,(
    ! [V_k_2,V_A_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),V_A_2),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_k_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k_2),hAPP(c_Finite__Set_Ocard(tc_Nat_Onat),V_A_2)))))
     => V_A_2 = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_k_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k_2),hAPP(c_Finite__Set_Ocard(tc_Nat_Onat),V_A_2))) ) )).

fof(fact_image__uminus__atLeastLessThan,axiom,(
    ! [V_y_2,V_x_2,T_b] :
      ( class_Groups_Oordered__ab__group__add(T_b)
     => hAPP(c_Set_Oimage(T_b,T_b,c_Groups_Ouminus__class_Ouminus(T_b)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_x_2),V_y_2)) = c_SetInterval_Oord__class_OgreaterThanAtMost(T_b,hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_y_2),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_x_2)) ) )).

fof(fact_image__uminus__greaterThanAtMost,axiom,(
    ! [V_y_2,V_x_2,T_b] :
      ( class_Groups_Oordered__ab__group__add(T_b)
     => hAPP(c_Set_Oimage(T_b,T_b,c_Groups_Ouminus__class_Ouminus(T_b)),c_SetInterval_Oord__class_OgreaterThanAtMost(T_b,V_x_2,V_y_2)) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_y_2)),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_x_2)) ) )).

fof(fact_setsum__shift__lb__Suc0__0__upt,axiom,(
    ! [V_k_2,V_f_2,T_b] :
      ( class_Groups_Ocomm__monoid__add(T_b)
     => ( hAPP(V_f_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = c_Groups_Ozero__class_Ozero(T_b)
       => hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),V_k_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_k_2)) ) ) )).

fof(fact_setsum__head__upt__Suc,axiom,(
    ! [V_f_2,V_n_2,V_m_2,T_b] :
      ( class_Groups_Ocomm__monoid__add(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2))
       => hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2),V_n_2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(V_f_2,V_m_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m_2)),V_n_2))) ) ) )).

fof(fact_card__atLeastZeroLessThan__int,axiom,(
    ! [V_u_2] : hAPP(c_Finite__Set_Ocard(tc_Int_Oint),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_u_2)) = hAPP(c_Int_Onat,V_u_2) )).

fof(fact_card__atLeastLessThan__int,axiom,(
    ! [V_u_2,V_l_2] : hAPP(c_Finite__Set_Ocard(tc_Int_Oint),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,V_l_2),V_u_2)) = hAPP(c_Int_Onat,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_u_2),V_l_2)) )).

fof(fact_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,(
    ! [V_u_2,V_l_2] : hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_l_2),c_Groups_Oone__class_Oone(tc_Int_Oint))),V_u_2) = c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Int_Oint,V_l_2,V_u_2) )).

fof(fact_atLeastLessThanSuc,axiom,(
    ! [V_n_2,V_m_2] :
      ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2))
       => hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2),hAPP(c_Nat_OSuc,V_n_2)) = hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),V_n_2),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2),V_n_2)) )
      & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2))
       => hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2),hAPP(c_Nat_OSuc,V_n_2)) = c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) ) ) )).

fof(fact_setsum__op__ivl__Suc,axiom,(
    ! [V_f_2,V_m_2,V_n_2,T_b] :
      ( class_Groups_Ocomm__monoid__add(T_b)
     => ( ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),V_m_2))
         => hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2),hAPP(c_Nat_OSuc,V_n_2))) = c_Groups_Ozero__class_Ozero(T_b) )
        & ( ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),V_m_2))
         => hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2),hAPP(c_Nat_OSuc,V_n_2))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2),V_n_2))),hAPP(V_f_2,V_n_2)) ) ) ) )).

fof(fact_ivl__disj__un_I15_J,axiom,(
    ! [V_u_2,V_m_2,V_l_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_l_2),V_m_2))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_m_2),V_u_2))
         => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanLessThan(T_b,V_l_2,V_m_2)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_m_2),V_u_2)) = c_SetInterval_Oord__class_OgreaterThanLessThan(T_b,V_l_2,V_u_2) ) ) ) )).

fof(fact_Image__singleton__iff,axiom,(
    ! [V_a_2,V_r_2,T_c,V_b_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),hAPP(c_Relation_OImage(T_c,T_b,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))))
    <=> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_b),V_a_2),V_b_2)),V_r_2)) ) )).

fof(fact_image__atLeastZeroLessThan__int,axiom,(
    ! [V_u_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_u_2))
     => hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_u_2) = hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),hAPP(c_Int_Onat,V_u_2))) ) )).

fof(fact_ivl__disj__un_I6_J,axiom,(
    ! [V_u_2,V_l_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_l_2),V_u_2))
       => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_l_2),V_u_2)),hAPP(hAPP(c_Set_Oinsert(T_b),V_u_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) = c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_l_2,V_u_2) ) ) )).

fof(fact_ivl__disj__un_I3_J,axiom,(
    ! [V_u_2,V_l_2,T_b] :
      ( class_Orderings_Olinorder(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_l_2),V_u_2))
       => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_b),V_l_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))),c_SetInterval_Oord__class_OgreaterThanLessThan(T_b,V_l_2,V_u_2)) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_l_2),V_u_2) ) ) )).

fof(fact_of__int,axiom,(
    ! [V_j_2,V_i_2,T_b] :
      ( class_Rings_Oring__1(T_b)
     => hAPP(c_Int_Oring__1__class_Oof__int(T_b),hAPP(c_Int_OAbs__Integ,hAPP(c_Relation_OImage(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Int_Ointrel),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_i_2),V_j_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)))))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_b),V_i_2)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_b),V_j_2)) ) )).

fof(fact_One__int__def,axiom,(
    c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(c_Int_OAbs__Integ,hAPP(c_Relation_OImage(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Int_Ointrel),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))))) )).

fof(fact_mult,axiom,(
    ! [V_v_2,V_u_2,V_y_2,V_x_2] : hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_OAbs__Integ,hAPP(c_Relation_OImage(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Int_Ointrel),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_x_2),V_y_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)))))),hAPP(c_Int_OAbs__Integ,hAPP(c_Relation_OImage(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Int_Ointrel),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_u_2),V_v_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)))))) = hAPP(c_Int_OAbs__Integ,hAPP(c_Relation_OImage(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Int_Ointrel),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_x_2),V_u_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_y_2),V_v_2))),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_x_2),V_v_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_y_2),V_u_2)))),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))))) )).

fof(fact_le,axiom,(
    ! [V_v_2,V_u_2,V_y_2,V_x_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Int_OAbs__Integ,hAPP(c_Relation_OImage(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Int_Ointrel),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_x_2),V_y_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)))))),hAPP(c_Int_OAbs__Integ,hAPP(c_Relation_OImage(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Int_Ointrel),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_u_2),V_v_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)))))))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_x_2),V_v_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_u_2),V_y_2))) ) )).

fof(fact_minus,axiom,(
    ! [V_y_2,V_x_2] : hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Int_OAbs__Integ,hAPP(c_Relation_OImage(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Int_Ointrel),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_x_2),V_y_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)))))) = hAPP(c_Int_OAbs__Integ,hAPP(c_Relation_OImage(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Int_Ointrel),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_y_2),V_x_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))))) )).

fof(fact_Zero__int__def,axiom,(
    c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(c_Int_OAbs__Integ,hAPP(c_Relation_OImage(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Int_Ointrel),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))))) )).

fof(fact_int__def,axiom,(
    ! [V_m_2] : hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m_2) = hAPP(c_Int_OAbs__Integ,hAPP(c_Relation_OImage(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Int_Ointrel),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_m_2),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))))) )).

fof(fact_add,axiom,(
    ! [V_v_2,V_u_2,V_y_2,V_x_2] : hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Int_OAbs__Integ,hAPP(c_Relation_OImage(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Int_Ointrel),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_x_2),V_y_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)))))),hAPP(c_Int_OAbs__Integ,hAPP(c_Relation_OImage(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Int_Ointrel),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_u_2),V_v_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)))))) = hAPP(c_Int_OAbs__Integ,hAPP(c_Relation_OImage(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Int_Ointrel),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_x_2),V_u_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_y_2),V_v_2))),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))))) )).

fof(fact_nat,axiom,(
    ! [V_y_2,V_x_2] : hAPP(c_Int_Onat,hAPP(c_Int_OAbs__Integ,hAPP(c_Relation_OImage(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Int_Ointrel),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_x_2),V_y_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)))))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_x_2),V_y_2) )).

fof(fact_less,axiom,(
    ! [V_v_2,V_u_2,V_y_2,V_x_2] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_OAbs__Integ,hAPP(c_Relation_OImage(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Int_Ointrel),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_x_2),V_y_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)))))),hAPP(c_Int_OAbs__Integ,hAPP(c_Relation_OImage(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Int_Ointrel),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_u_2),V_v_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)))))))
    <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_x_2),V_v_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_u_2),V_y_2))) ) )).

fof(fact_eq__Abs__Integ,axiom,(
    ! [V_z_2] :
      ~ ! [B_x,B_y] : V_z_2 != hAPP(c_Int_OAbs__Integ,hAPP(c_Relation_OImage(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Int_Ointrel),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),B_x),B_y)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))))) )).

fof(fact_ImageE,axiom,(
    ! [V_A_2,V_r_2,T_c,V_b_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),hAPP(c_Relation_OImage(T_c,T_b,V_r_2),V_A_2)))
     => ~ ! [B_x] :
            ( hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_b),B_x),V_b_2)),V_r_2))
           => ~ hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)) ) ) )).

fof(fact_fold__Un__disjoint,axiom,(
    ! [V_g_2,V_B_2,V_A_2,T_b,T_c] :
      ( class_Groups_Ocomm__monoid__mult(T_c)
     => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))
       => ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2))
         => ( hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
           => hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_c,T_b,c_Groups_Otimes__class_Otimes(T_c)),V_g_2),c_Groups_Oone__class_Oone(T_c)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_c,T_b,c_Groups_Otimes__class_Otimes(T_c)),V_g_2),c_Groups_Oone__class_Oone(T_c)),V_A_2)),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_c,T_b,c_Groups_Otimes__class_Otimes(T_c)),V_g_2),c_Groups_Oone__class_Oone(T_c)),V_B_2)) ) ) ) ) )).

fof(fact_subset__equiv__class,axiom,(
    ! [V_a_2,V_b_2,V_r_2,V_A_2,T_b] :
      ( c_Equiv__Relations_Oequiv(T_b,V_A_2,V_r_2)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_b,T_b,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_b_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))),hAPP(c_Relation_OImage(T_b,T_b,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))))
       => ( hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),V_A_2))
         => hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_b_2)),V_r_2)) ) ) ) )).

fof(fact_inf1E,axiom,(
    ! [V_x_2,V_B_2,V_A_2,T_b] :
      ( hBOOL(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2),V_x_2))
     => ~ ( hBOOL(hAPP(V_A_2,V_x_2))
         => ~ hBOOL(hAPP(V_B_2,V_x_2)) ) ) )).

fof(fact_inf1I,axiom,(
    ! [T_b,V_B_2,V_x_2,V_A_2] :
      ( hBOOL(hAPP(V_A_2,V_x_2))
     => ( hBOOL(hAPP(V_B_2,V_x_2))
       => hBOOL(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2),V_x_2)) ) ) )).

fof(fact_IntI,axiom,(
    ! [V_B_2,V_A_2,V_c_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_A_2))
     => ( hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_B_2))
       => hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) ) ) )).

fof(fact_IntE,axiom,(
    ! [V_B_2,V_A_2,V_c_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)))
     => ~ ( hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_A_2))
         => ~ hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_B_2)) ) ) )).

fof(fact_finite__Int,axiom,(
    ! [V_Ga_2,V_F_2,T_b] :
      ( ( hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_F_2))
        | hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_Ga_2)) )
     => hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_F_2),V_Ga_2))) ) )).

fof(fact_insert__inter__insert,axiom,(
    ! [V_B_2,V_A_2,V_a_2,T_b] : hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2)),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_B_2)) = hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) )).

fof(fact_inf__bot__right,axiom,(
    ! [V_x,T_a] :
      ( class_Lattices_Obounded__lattice__bot(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),c_Orderings_Obot__class_Obot(T_a)) = c_Orderings_Obot__class_Obot(T_a) ) )).

fof(fact_inf__bot__left,axiom,(
    ! [V_x,T_a] :
      ( class_Lattices_Obounded__lattice__bot(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),c_Orderings_Obot__class_Obot(T_a)),V_x) = c_Orderings_Obot__class_Obot(T_a) ) )).

fof(fact_disjoint__iff__not__equal,axiom,(
    ! [V_B_2,V_A_2,T_b] :
      ( hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))
    <=> ! [B_x] :
          ( hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2))
         => ! [B_xa] :
              ( hBOOL(hAPP(hAPP(c_member(T_b),B_xa),V_B_2))
             => B_x != B_xa ) ) ) )).

fof(fact_Int__empty__right,axiom,(
    ! [V_A_2,T_b] : hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) )).

fof(fact_Int__empty__left,axiom,(
    ! [V_B_2,T_b] : hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))),V_B_2) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) )).

fof(fact_Nat__Transfer_Otransfer__int__nat__set__function__closures_I3_J,axiom,(
    ! [V_B_2,V_A_2] :
      ( c_Nat__Transfer_Onat__set(V_A_2)
     => ( c_Nat__Transfer_Onat__set(V_B_2)
       => c_Nat__Transfer_Onat__set(hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Int_Oint,tc_HOL_Obool)),V_A_2),V_B_2)) ) ) )).

fof(fact_inf__min,axiom,(
    ! [T_b] :
      ( ( class_Lattices_Osemilattice__inf(T_b)
        & class_Orderings_Olinorder(T_b) )
     => c_Lattices_Osemilattice__inf__class_Oinf(T_b) = c_Orderings_Oord__class_Omin(T_b) ) )).

fof(fact_inf__apply,axiom,(
    ! [V_x_2,V_g_2,V_f_2,T_c,T_b] :
      ( class_Lattices_Olattice(T_b)
     => hAPP(hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,T_b)),V_f_2),V_g_2),V_x_2) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_b),hAPP(V_f_2,V_x_2)),hAPP(V_g_2,V_x_2)) ) )).

fof(fact_inf__assoc,axiom,(
    ! [V_z,V_y,V_x,T_a] :
      ( class_Lattices_Osemilattice__inf(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y)),V_z) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_y),V_z)) ) )).

fof(fact_inf__sup__aci_I2_J,axiom,(
    ! [V_z,V_y,V_x,T_a] :
      ( class_Lattices_Olattice(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y)),V_z) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_y),V_z)) ) )).

fof(fact_inf_Oassoc,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Lattices_Osemilattice__inf(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),V_b)),V_c) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_b),V_c)) ) )).

fof(fact_inf__left__commute,axiom,(
    ! [V_z,V_y,V_x,T_a] :
      ( class_Lattices_Osemilattice__inf(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_y),V_z)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_y),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_z)) ) )).

fof(fact_inf__sup__aci_I3_J,axiom,(
    ! [V_z,V_y,V_x,T_a] :
      ( class_Lattices_Olattice(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_y),V_z)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_y),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_z)) ) )).

fof(fact_inf_Oleft__commute,axiom,(
    ! [V_c,V_a,V_b,T_a] :
      ( class_Lattices_Osemilattice__inf(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_b),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),V_c)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_b),V_c)) ) )).

fof(fact_inf__left__idem,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Lattices_Osemilattice__inf(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y) ) )).

fof(fact_inf__sup__aci_I4_J,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Lattices_Olattice(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y) ) )).

fof(fact_inf_Oleft__idem,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Lattices_Osemilattice__inf(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),V_b)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),V_b) ) )).

fof(fact_inf__commute,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Lattices_Osemilattice__inf(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_y),V_x) ) )).

fof(fact_inf__sup__aci_I1_J,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Lattices_Olattice(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_y),V_x) ) )).

fof(fact_inf_Ocommute,axiom,(
    ! [V_b,V_a,T_a] :
      ( class_Lattices_Osemilattice__inf(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),V_b) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_b),V_a) ) )).

fof(fact_inf__idem,axiom,(
    ! [V_x,T_a] :
      ( class_Lattices_Osemilattice__inf(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_x) = V_x ) )).

fof(fact_inf_Oidem,axiom,(
    ! [V_a,T_a] :
      ( class_Lattices_Osemilattice__inf(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),V_a) = V_a ) )).

fof(fact_Int__assoc,axiom,(
    ! [V_C_2,V_B_2,V_A_2,T_b] : hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),V_C_2) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2)) )).

fof(fact_Int__left__commute,axiom,(
    ! [V_C_2,V_B_2,V_A_2,T_b] : hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_B_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_C_2)) )).

fof(fact_Int__left__absorb,axiom,(
    ! [V_B_2,V_A_2,T_b] : hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) )).

fof(fact_Int__commute,axiom,(
    ! [V_B_2,V_A_2,T_b] : hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2) )).

fof(fact_Int__absorb,axiom,(
    ! [V_A_2,T_b] : hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_A_2) = V_A_2 )).

fof(fact_inf1D2,axiom,(
    ! [V_x_2,V_B_2,V_A_2,T_b] :
      ( hBOOL(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2),V_x_2))
     => hBOOL(hAPP(V_B_2,V_x_2)) ) )).

fof(fact_inf1D1,axiom,(
    ! [V_x_2,V_B_2,V_A_2,T_b] :
      ( hBOOL(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2),V_x_2))
     => hBOOL(hAPP(V_A_2,V_x_2)) ) )).

fof(fact_Inf__fin_Oidem,axiom,(
    ! [V_x,T_a] :
      ( class_Lattices_Olattice(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_x) = V_x ) )).

fof(fact_Pow__Int__eq,axiom,(
    ! [V_B_2,V_A_2,T_b] : hAPP(c_Set_OPow(T_b),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)),hAPP(c_Set_OPow(T_b),V_A_2)),hAPP(c_Set_OPow(T_b),V_B_2)) )).

fof(fact_Diff__Int__distrib,axiom,(
    ! [V_B_2,V_A_2,V_C_2,T_b] : hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_A_2)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_B_2)) )).

fof(fact_Int__Diff,axiom,(
    ! [V_C_2,V_B_2,V_A_2,T_b] : hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),V_C_2) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2)) )).

fof(fact_Diff__Int__distrib2,axiom,(
    ! [V_C_2,V_B_2,V_A_2,T_b] : hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),V_C_2) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_C_2)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2)) )).

fof(fact_Diff__Int2,axiom,(
    ! [V_B_2,V_C_2,V_A_2,T_b] : hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_C_2)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_C_2)),V_B_2) )).

fof(fact_less__infI1,axiom,(
    ! [V_b,V_x,V_a,T_a] :
      ( class_Lattices_Osemilattice__inf(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_x))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),V_b)),V_x)) ) ) )).

fof(fact_less__infI2,axiom,(
    ! [V_a,V_x,V_b,T_a] :
      ( class_Lattices_Osemilattice__inf(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),V_x))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),V_b)),V_x)) ) ) )).

fof(fact_IntD2,axiom,(
    ! [V_B_2,V_A_2,V_c_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)))
     => hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_B_2)) ) )).

fof(fact_IntD1,axiom,(
    ! [V_B_2,V_A_2,V_c_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)))
     => hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_A_2)) ) )).

fof(fact_Int__iff,axiom,(
    ! [V_B_2,V_A_2,V_c_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)))
    <=> ( hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_A_2))
        & hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_B_2)) ) ) )).

fof(fact_Int__mono,axiom,(
    ! [V_D_2,V_B_2,V_C_2,V_A_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_C_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_D_2))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_D_2))) ) ) )).

fof(fact_Int__greatest,axiom,(
    ! [V_B_2,V_A_2,V_C_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_A_2))
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_B_2))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) ) ) )).

fof(fact_Int__absorb1,axiom,(
    ! [V_A_2,V_B_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2))
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) = V_B_2 ) )).

fof(fact_Int__absorb2,axiom,(
    ! [V_B_2,V_A_2,T_b] :
      ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) = V_A_2 ) )).

fof(fact_Int__lower2,axiom,(
    ! [V_B_2,V_A_2,T_b] : hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),V_B_2)) )).

fof(fact_Int__lower1,axiom,(
    ! [V_B_2,V_A_2,T_b] : hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),V_A_2)) )).

fof(fact_inf__sup__ord_I1_J,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Lattices_Olattice(T_a)
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y)),V_x)) ) )).

fof(fact_inf__le1,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Lattices_Osemilattice__inf(T_a)
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y)),V_x)) ) )).

fof(fact_inf__sup__ord_I2_J,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Lattices_Olattice(T_a)
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y)),V_y)) ) )).

fof(fact_inf__le2,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Lattices_Osemilattice__inf(T_a)
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y)),V_y)) ) )).

fof(fact_le__iff__inf,axiom,(
    ! [V_y_2,V_x_2,T_b] :
      ( class_Lattices_Osemilattice__inf(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),V_y_2))
      <=> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_b),V_x_2),V_y_2) = V_x_2 ) ) )).

fof(fact_le__inf__iff,axiom,(
    ! [V_z_2,V_y_2,V_x_2,T_b] :
      ( class_Lattices_Osemilattice__inf(T_b)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_b),V_y_2),V_z_2)))
      <=> ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),V_y_2))
          & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),V_z_2)) ) ) ) )).

fof(fact_le__infI1,axiom,(
    ! [V_b,V_x,V_a,T_a] :
      ( class_Lattices_Osemilattice__inf(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_x))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),V_b)),V_x)) ) ) )).

fof(fact_le__infI2,axiom,(
    ! [V_a,V_x,V_b,T_a] :
      ( class_Lattices_Osemilattice__inf(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),V_x))
       => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),V_b)),V_x)) ) ) )).

fof(fact_inf__absorb1,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Lattices_Osemilattice__inf(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y))
       => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y) = V_x ) ) )).

fof(fact_inf__absorb2,axiom,(
    ! [V_x,V_y,T_a] :
      ( class_Lattices_Osemilattice__inf(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),V_x))
       => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y) = V_y ) ) )).

fof(fact_le__infI,axiom,(
    ! [V_b,V_a,V_x,T_a] :
      ( class_Lattices_Osemilattice__inf(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_a))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_b))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),V_b))) ) ) ) )).

fof(fact_inf__greatest,axiom,(
    ! [V_z,V_y,V_x,T_a] :
      ( class_Lattices_Osemilattice__inf(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_z))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_y),V_z))) ) ) ) )).

fof(fact_inf__mono,axiom,(
    ! [V_d,V_b,V_c,V_a,T_a] :
      ( class_Lattices_Osemilattice__inf(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_c))
       => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),V_d))
         => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),V_b)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_c),V_d))) ) ) ) )).

fof(fact_le__infE,axiom,(
    ! [V_b,V_a,V_x,T_a] :
      ( class_Lattices_Osemilattice__inf(T_a)
     => ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),V_b)))
       => ~ ( hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_a))
           => ~ hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_b)) ) ) ) )).

fof(fact_fun__left__comm__idem__inf,axiom,(
    ! [T_b] :
      ( class_Lattices_Osemilattice__inf(T_b)
     => c_Finite__Set_Ofun__left__comm__idem(T_b,T_b,c_Lattices_Osemilattice__inf__class_Oinf(T_b)) ) )).

fof(fact_sup__inf__distrib2,axiom,(
    ! [V_x,V_z,V_y,T_a] :
      ( class_Lattices_Odistrib__lattice(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_y),V_z)),V_x) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_y),V_x)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_z),V_x)) ) )).

fof(fact_inf__sup__distrib2,axiom,(
    ! [V_x,V_z,V_y,T_a] :
      ( class_Lattices_Odistrib__lattice(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_y),V_z)),V_x) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_y),V_x)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_z),V_x)) ) )).

fof(fact_sup__inf__distrib1,axiom,(
    ! [V_z,V_y,V_x,T_a] :
      ( class_Lattices_Odistrib__lattice(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_y),V_z)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_z)) ) )).

fof(fact_inf__sup__distrib1,axiom,(
    ! [V_z,V_y,V_x,T_a] :
      ( class_Lattices_Odistrib__lattice(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_y),V_z)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_z)) ) )).

fof(fact_sup__inf__absorb,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Lattices_Olattice(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y)) = V_x ) )).

fof(fact_inf__sup__absorb,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Lattices_Olattice(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y)) = V_x ) )).

fof(fact_Un__Int__crazy,axiom,(
    ! [V_C_2,V_B_2,V_A_2,T_b] : hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2))),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_A_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2))),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_A_2)) )).

fof(fact_Un__Int__distrib2,axiom,(
    ! [V_A_2,V_C_2,V_B_2,T_b] : hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2)),V_A_2) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_A_2)) )).

fof(fact_Int__Un__distrib2,axiom,(
    ! [V_A_2,V_C_2,V_B_2,T_b] : hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2)),V_A_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_A_2)) )).

fof(fact_Un__Int__distrib,axiom,(
    ! [V_C_2,V_B_2,V_A_2,T_b] : hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_C_2)) )).

fof(fact_Int__Un__distrib,axiom,(
    ! [V_C_2,V_B_2,V_A_2,T_b] : hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_C_2)) )).

fof(fact_distrib__sup__le,axiom,(
    ! [V_z,V_y,V_x,T_a] :
      ( class_Lattices_Olattice(T_a)
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_y),V_z))),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_z)))) ) )).

fof(fact_distrib__inf__le,axiom,(
    ! [V_z,V_y,V_x,T_a] :
      ( class_Lattices_Olattice(T_a)
     => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_z))),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_y),V_z)))) ) )).

fof(fact_inf__compl__bot,axiom,(
    ! [V_x,T_a] :
      ( class_Lattices_Oboolean__algebra(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_x)) = c_Orderings_Obot__class_Obot(T_a) ) )).

fof(fact_compl__inf__bot,axiom,(
    ! [V_x,T_a] :
      ( class_Lattices_Oboolean__algebra(T_a)
     => hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_x)),V_x) = c_Orderings_Obot__class_Obot(T_a) ) )).

fof(fact_diff__eq,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Lattices_Oboolea