TPTP Problem File: SCT115+1.p

View Solutions - Solve Problem

%------------------------------------------------------------------------------
% File     : SCT115+1 : TPTP v7.4.0. Released v5.2.0.
% Domain   : Social Choice Theory
% Problem  : Arrow's Impossibility Theorem 432712, 500 axioms selected
% Version  : Especial.
% English  :

% Refs     : [BN10]  Boehme & Nipkow (2010), Sledgehammer: Judgement Day
%          : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% Source   : [Bla11]
% Names    : arrow_432712.500.p [Bla11]

% Status   : Theorem
% Rating   : 0.78 v7.4.0, 0.60 v7.3.0, 0.66 v7.2.0, 0.62 v7.1.0, 0.70 v7.0.0, 0.67 v6.4.0, 0.69 v6.3.0, 0.71 v6.2.0, 0.68 v6.1.0, 0.77 v6.0.0, 0.74 v5.5.0, 0.85 v5.4.0, 0.93 v5.2.0
% Syntax   : Number of formulae    :  528 ( 134 unit)
%            Number of atoms       : 1361 ( 300 equality)
%            Maximal formula depth :   19 (   6 average)
%            Number of connectives :  994 ( 161   ~;  35   |;  87   &)
%                                         ( 111 <=>; 600  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of predicates  :   36 (   0 propositional; 1-6 arity)
%            Number of functors    :   48 (   9 constant; 0-6 arity)
%            Number of variables   : 1842 (   0 sgn;1825   !;  17   ?)
%            Maximal term depth    :    6 (   2 average)
% SPC      : FOF_THM_RFO_SEQ

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2011-03-01 11:09:16
%          : Renamed from SWW117+1 
%------------------------------------------------------------------------------
%----Relevant facts (494)
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_equalityCE,axiom,(
    ! [V_c_2,T_a,V_B_2,V_A_2] :
      ( V_A_2 = V_B_2
     => ( ( c_member(T_a,V_c_2,V_A_2)
         => ~ c_member(T_a,V_c_2,V_B_2) )
       => ~ ( ~ c_member(T_a,V_c_2,V_A_2)
           => c_member(T_a,V_c_2,V_B_2) ) ) ) )).

fof(fact_in__mkbot,axiom,(
    ! [V_z_2,V_La_2,V_ya_2,V_xa_2] :
      ( c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,V_xa_2,V_ya_2),c_Arrow__Order__Mirabelle_Omkbot(V_La_2,V_z_2))
    <=> ( V_ya_2 != V_z_2
        & ( V_xa_2 = V_z_2
         => V_xa_2 != V_ya_2 )
        & ( V_xa_2 != V_z_2
         => c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,V_xa_2,V_ya_2),V_La_2) ) ) ) )).

fof(fact_in__mktop,axiom,(
    ! [V_z_2,V_La_2,V_ya_2,V_xa_2] :
      ( c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,V_xa_2,V_ya_2),c_Arrow__Order__Mirabelle_Omktop(V_La_2,V_z_2))
    <=> ( V_xa_2 != V_z_2
        & ( V_ya_2 = V_z_2
         => V_xa_2 != V_ya_2 )
        & ( V_ya_2 != V_z_2
         => c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,V_xa_2,V_ya_2),V_La_2) ) ) ) )).

fof(fact_in__rel__def,axiom,(
    ! [V_ya_2,V_xa_2,V_R_2,T_b,T_a] :
      ( hBOOL(hAPP(hAPP(c_FunDef_Oin__rel(T_a,T_b,V_R_2),V_xa_2),V_ya_2))
    <=> c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_xa_2,V_ya_2),V_R_2) ) )).

fof(fact_same__fstI,axiom,(
    ! [T_a,V_R_2,V_ya_2,V_y_H_2,T_b,V_xa_2,V_P_2] :
      ( hBOOL(hAPP(V_P_2,V_xa_2))
     => ( c_member(tc_prod(T_b,T_b),c_Product__Type_OPair(T_b,T_b,V_y_H_2,V_ya_2),hAPP(V_R_2,V_xa_2))
       => c_member(tc_prod(tc_prod(T_a,T_b),tc_prod(T_a,T_b)),c_Product__Type_OPair(tc_prod(T_a,T_b),tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_xa_2,V_y_H_2),c_Product__Type_OPair(T_a,T_b,V_xa_2,V_ya_2)),c_Recdef_Osame__fst(T_a,T_b,V_P_2,V_R_2)) ) ) )).

fof(fact_Nitpick_Orefl_H__def,axiom,(
    ! [V_r_2,T_a] :
      ( c_Nitpick_Orefl_H(T_a,V_r_2)
    <=> ! [B_x] : c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_x,B_x),V_r_2) ) )).

fof(fact_split__paired__All,axiom,(
    ! [T_b,T_a,V_P_2] :
      ( ! [B_x1] : hBOOL(hAPP(V_P_2,B_x1))
    <=> ! [B_a,B_b] : hBOOL(hAPP(V_P_2,c_Product__Type_OPair(T_a,T_b,B_a,B_b))) ) )).

fof(fact_Pair__eq,axiom,(
    ! [V_b_H_2,V_a_H_2,V_b_2,V_a_2,T_b,T_a] :
      ( c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2) = c_Product__Type_OPair(T_a,T_b,V_a_H_2,V_b_H_2)
    <=> ( V_a_2 = V_a_H_2
        & V_b_2 = V_b_H_2 ) ) )).

fof(fact_Pair__inject,axiom,(
    ! [V_b_H,V_a_H,V_b,V_a,T_b,T_a] :
      ( c_Product__Type_OPair(T_a,T_b,V_a,V_b) = 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_mem__def,axiom,(
    ! [V_A_2,V_xa_2,T_a] :
      ( c_member(T_a,V_xa_2,V_A_2)
    <=> hBOOL(hAPP(V_A_2,V_xa_2)) ) )).

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

fof(fact_eqelem__imp__iff,axiom,(
    ! [V_A_2,T_a,V_ya_2,V_xa_2] :
      ( V_xa_2 = V_ya_2
     => ( c_member(T_a,V_xa_2,V_A_2)
      <=> c_member(T_a,V_ya_2,V_A_2) ) ) )).

fof(fact_eqset__imp__iff,axiom,(
    ! [V_xa_2,T_a,V_B_2,V_A_2] :
      ( V_A_2 = V_B_2
     => ( c_member(T_a,V_xa_2,V_A_2)
      <=> c_member(T_a,V_xa_2,V_B_2) ) ) )).

fof(fact_prod_Orecs,axiom,(
    ! [V_b_2,V_a_2,V_f1_2,T_a,T_c,T_b] : c_Product__Type_Oprod_Oprod__rec(T_b,T_c,T_a,V_f1_2,c_Product__Type_OPair(T_b,T_c,V_a_2,V_b_2)) = hAPP(hAPP(V_f1_2,V_a_2),V_b_2) )).

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_b,T_a] :
      ( c_member(tc_prod(tc_prod(T_a,T_b),tc_prod(T_a,T_b)),c_Product__Type_OPair(tc_prod(T_a,T_b),tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2),c_Product__Type_OPair(T_a,T_b,V_a_H_2,V_b_H_2)),c_Wellfounded_Olex__prod(T_a,T_b,V_r_2,V_s_2))
    <=> ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_a_H_2),V_r_2)
        | ( V_a_2 = V_a_H_2
          & c_member(tc_prod(T_b,T_b),c_Product__Type_OPair(T_b,T_b,V_b_2,V_b_H_2),V_s_2) ) ) ) )).

fof(fact_internal__split__conv,axiom,(
    ! [V_b_2,V_a_2,V_c_2,T_a,T_c,T_b] : c_Product__Type_Ointernal__split(T_b,T_c,T_a,V_c_2,c_Product__Type_OPair(T_b,T_c,V_a_2,V_b_2)) = hAPP(hAPP(V_c_2,V_a_2),V_b_2) )).

fof(fact_Lin__irrefl,axiom,(
    ! [V_b_2,V_a_2,V_La_2] :
      ( c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),V_La_2,c_Arrow__Order__Mirabelle_OLin)
     => ( c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,V_a_2,V_b_2),V_La_2)
       => ~ c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,V_b_2,V_a_2),V_La_2) ) ) )).

fof(fact_notin__Lin__iff,axiom,(
    ! [V_ya_2,V_xa_2,V_La_2] :
      ( c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),V_La_2,c_Arrow__Order__Mirabelle_OLin)
     => ( V_xa_2 != V_ya_2
       => ( ~ c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,V_xa_2,V_ya_2),V_La_2)
        <=> c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,V_ya_2,V_xa_2),V_La_2) ) ) ) )).

fof(fact_wfrec__rel_Oequations,axiom,(
    ! [V_a2_2,V_a1_2,V_F_2,V_R_2,T_b,T_a] :
      ( c_Recdef_Owfrec__rel(T_a,T_b,V_R_2,V_F_2,V_a1_2,hAPP(hAPP(V_F_2,V_a2_2),V_a1_2))
    <=> ? [B_g] :
          ( hAPP(hAPP(V_F_2,V_a2_2),V_a1_2) = hAPP(hAPP(V_F_2,B_g),V_a1_2)
          & ! [B_z] :
              ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_z,V_a1_2),V_R_2)
             => c_Recdef_Owfrec__rel(T_a,T_b,V_R_2,V_F_2,B_z,hAPP(B_g,B_z)) ) ) ) )).

fof(fact_mkbot__Lin,axiom,(
    ! [V_xa_2,V_La_2] :
      ( c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),V_La_2,c_Arrow__Order__Mirabelle_OLin)
     => c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),c_Arrow__Order__Mirabelle_Omkbot(V_La_2,V_xa_2),c_Arrow__Order__Mirabelle_OLin) ) )).

fof(fact_mktop__Lin,axiom,(
    ! [V_xa_2,V_La_2] :
      ( c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),V_La_2,c_Arrow__Order__Mirabelle_OLin)
     => c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),c_Arrow__Order__Mirabelle_Omktop(V_La_2,V_xa_2),c_Arrow__Order__Mirabelle_OLin) ) )).

fof(fact_curryI,axiom,(
    ! [V_b_2,V_a_2,T_b,T_a,V_f_2] :
      ( hBOOL(hAPP(V_f_2,c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2)))
     => hBOOL(c_Product__Type_Ocurry(T_a,T_b,tc_HOL_Obool,V_f_2,V_a_2,V_b_2)) ) )).

fof(fact_cut__apply,axiom,(
    ! [V_f_2,T_b,V_r_2,V_a_2,V_xa_2,T_a] :
      ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_a_2),V_r_2)
     => hAPP(c_Recdef_Ocut(T_a,T_b,V_f_2,V_r_2,V_a_2),V_xa_2) = hAPP(V_f_2,V_xa_2) ) )).

fof(fact_cuts__eq,axiom,(
    ! [V_g_2,V_xa_2,V_r_2,V_f_2,T_b,T_a] :
      ( c_Recdef_Ocut(T_a,T_b,V_f_2,V_r_2,V_xa_2) = c_Recdef_Ocut(T_a,T_b,V_g_2,V_r_2,V_xa_2)
    <=> ! [B_y] :
          ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_y,V_xa_2),V_r_2)
         => hAPP(V_f_2,B_y) = hAPP(V_g_2,B_y) ) ) )).

fof(fact_curryD,axiom,(
    ! [V_b_2,V_a_2,V_f_2,T_b,T_a] :
      ( hBOOL(c_Product__Type_Ocurry(T_a,T_b,tc_HOL_Obool,V_f_2,V_a_2,V_b_2))
     => hBOOL(hAPP(V_f_2,c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2))) ) )).

fof(fact_curryE,axiom,(
    ! [V_b_2,V_a_2,V_f_2,T_b,T_a] :
      ( hBOOL(c_Product__Type_Ocurry(T_a,T_b,tc_HOL_Obool,V_f_2,V_a_2,V_b_2))
     => hBOOL(hAPP(V_f_2,c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2))) ) )).

fof(fact_curry__conv,axiom,(
    ! [V_b_2,V_a_2,V_f_2,T_a,T_c,T_b] : c_Product__Type_Ocurry(T_b,T_c,T_a,V_f_2,V_a_2,V_b_2) = hAPP(V_f_2,c_Product__Type_OPair(T_b,T_c,V_a_2,V_b_2)) )).

fof(fact_tfl__cut__apply,axiom,(
    ! [T_b,V_a_2,V_xa_2,T_a,B_f,B_R] :
      ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_a_2),B_R)
     => hAPP(c_Recdef_Ocut(T_a,T_b,B_f,B_R,V_a_2),V_xa_2) = hAPP(B_f,V_xa_2) ) )).

fof(fact_in__above,axiom,(
    ! [V_ya_2,V_xa_2,V_La_2,V_b_2,V_a_2] :
      ( V_a_2 != V_b_2
     => ( c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),V_La_2,c_Arrow__Order__Mirabelle_OLin)
       => ( c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,V_xa_2,V_ya_2),c_Arrow__Order__Mirabelle_Oabove(V_La_2,V_a_2,V_b_2))
        <=> ( V_xa_2 != V_ya_2
            & ( V_xa_2 = V_b_2
             => c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,V_a_2,V_ya_2),V_La_2) )
            & ( V_xa_2 != V_b_2
             => ( ( V_ya_2 = V_b_2
                 => ( V_xa_2 = V_a_2
                    | c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,V_xa_2,V_a_2),V_La_2) ) )
                & ( V_ya_2 != V_b_2
                 => c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,V_xa_2,V_ya_2),V_La_2) ) ) ) ) ) ) ) )).

fof(fact_in__below,axiom,(
    ! [V_ya_2,V_xa_2,V_La_2,V_b_2,V_a_2] :
      ( V_a_2 != V_b_2
     => ( c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),V_La_2,c_Arrow__Order__Mirabelle_OLin)
       => ( c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,V_xa_2,V_ya_2),c_Arrow__Order__Mirabelle_Obelow(V_La_2,V_a_2,V_b_2))
        <=> ( V_xa_2 != V_ya_2
            & ( V_ya_2 = V_a_2
             => c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,V_xa_2,V_b_2),V_La_2) )
            & ( V_ya_2 != V_a_2
             => ( ( V_xa_2 = V_a_2
                 => ( V_ya_2 = V_b_2
                    | c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,V_b_2,V_ya_2),V_La_2) ) )
                & ( V_xa_2 != V_a_2
                 => c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,V_xa_2,V_ya_2),V_La_2) ) ) ) ) ) ) ) )).

fof(fact_wfrec__rel_Osimps,axiom,(
    ! [V_a2_2,V_a1_2,V_F_2,V_R_2,T_b,T_a] :
      ( c_Recdef_Owfrec__rel(T_a,T_b,V_R_2,V_F_2,V_a1_2,V_a2_2)
    <=> ? [B_g] :
          ( V_a2_2 = hAPP(hAPP(V_F_2,B_g),V_a1_2)
          & ! [B_z] :
              ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_z,V_a1_2),V_R_2)
             => c_Recdef_Owfrec__rel(T_a,T_b,V_R_2,V_F_2,B_z,hAPP(B_g,B_z)) ) ) ) )).

fof(fact_wfrec__rel_Ointros,axiom,(
    ! [V_g_2,V_F_2,T_b,V_R_2,V_xa_2,T_a] :
      ( ! [B_z] :
          ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_z,V_xa_2),V_R_2)
         => c_Recdef_Owfrec__rel(T_a,T_b,V_R_2,V_F_2,B_z,hAPP(V_g_2,B_z)) )
     => c_Recdef_Owfrec__rel(T_a,T_b,V_R_2,V_F_2,V_xa_2,hAPP(hAPP(V_F_2,V_g_2),V_xa_2)) ) )).

fof(fact_acc_Osimps,axiom,(
    ! [V_r_2,V_a_2,T_a] :
      ( c_member(T_a,V_a_2,c_Wellfounded_Oacc(T_a,V_r_2))
    <=> ! [B_x] :
          ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_x,V_a_2),V_r_2)
         => c_member(T_a,B_x,c_Wellfounded_Oacc(T_a,V_r_2)) ) ) )).

fof(fact_acc__downward,axiom,(
    ! [V_a_2,V_r_2,V_b_2,T_a] :
      ( c_member(T_a,V_b_2,c_Wellfounded_Oacc(T_a,V_r_2))
     => ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),V_r_2)
       => c_member(T_a,V_a_2,c_Wellfounded_Oacc(T_a,V_r_2)) ) ) )).

fof(fact_linear__alt,axiom,(
    ? [B_L] : c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),B_L,c_Arrow__Order__Mirabelle_OLin) )).

fof(fact_Nitpick_Owf__wfrec_H__def,axiom,(
    ! [V_xa_2,V_F_2,V_R_2,T_a,T_b] : c_Nitpick_Owf__wfrec_H(T_b,T_a,V_R_2,V_F_2,V_xa_2) = hAPP(hAPP(V_F_2,c_Recdef_Ocut(T_b,T_a,c_Nitpick_Owf__wfrec(T_b,T_a,V_R_2,V_F_2),V_R_2,V_xa_2)),V_xa_2) )).

fof(fact_eq__mem,axiom,(
    ! [V_ya_2,V_xa_2,T_a] :
      ( c_member(T_a,V_xa_2,c_fequal(V_ya_2))
    <=> V_xa_2 = V_ya_2 ) )).

fof(fact_SigmaI,axiom,(
    ! [V_B_2,V_b_2,T_b,V_A_2,V_a_2,T_a] :
      ( c_member(T_a,V_a_2,V_A_2)
     => ( c_member(T_b,V_b_2,hAPP(V_B_2,V_a_2))
       => c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2),c_Product__Type_OSigma(T_a,T_b,V_A_2,V_B_2)) ) ) )).

fof(fact_SigmaE2,axiom,(
    ! [V_B_2,V_A_2,V_b_2,V_a_2,T_b,T_a] :
      ( c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2),c_Product__Type_OSigma(T_a,T_b,V_A_2,V_B_2))
     => ~ ( c_member(T_a,V_a_2,V_A_2)
         => ~ c_member(T_b,V_b_2,hAPP(V_B_2,V_a_2)) ) ) )).

fof(fact_SigmaD2,axiom,(
    ! [V_B_2,V_A_2,V_b_2,V_a_2,T_b,T_a] :
      ( c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2),c_Product__Type_OSigma(T_a,T_b,V_A_2,V_B_2))
     => c_member(T_b,V_b_2,hAPP(V_B_2,V_a_2)) ) )).

fof(fact_SigmaD1,axiom,(
    ! [V_B_2,V_A_2,V_b_2,V_a_2,T_b,T_a] :
      ( c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2),c_Product__Type_OSigma(T_a,T_b,V_A_2,V_B_2))
     => c_member(T_a,V_a_2,V_A_2) ) )).

fof(fact_mem__Sigma__iff,axiom,(
    ! [V_B_2,V_A_2,V_b_2,V_a_2,T_b,T_a] :
      ( c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2),c_Product__Type_OSigma(T_a,T_b,V_A_2,V_B_2))
    <=> ( c_member(T_a,V_a_2,V_A_2)
        & c_member(T_b,V_b_2,hAPP(V_B_2,V_a_2)) ) ) )).

fof(fact_SigmaE,axiom,(
    ! [V_B_2,V_A_2,V_c_2,T_b,T_a] :
      ( c_member(tc_prod(T_a,T_b),V_c_2,c_Product__Type_OSigma(T_a,T_b,V_A_2,V_B_2))
     => ~ ! [B_x] :
            ( c_member(T_a,B_x,V_A_2)
           => ! [B_y] :
                ( c_member(T_b,B_y,hAPP(V_B_2,B_x))
               => V_c_2 != c_Product__Type_OPair(T_a,T_b,B_x,B_y) ) ) ) )).

fof(fact_not__acc__down,axiom,(
    ! [V_R_2,V_xa_2,T_a] :
      ( ~ c_member(T_a,V_xa_2,c_Wellfounded_Oacc(T_a,V_R_2))
     => ~ ! [B_z] :
            ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_z,V_xa_2),V_R_2)
           => c_member(T_a,B_z,c_Wellfounded_Oacc(T_a,V_R_2)) ) ) )).

fof(fact_acc_OaccI,axiom,(
    ! [V_r_2,V_xa_2,T_a] :
      ( ! [B_y] :
          ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_y,V_xa_2),V_r_2)
         => c_member(T_a,B_y,c_Wellfounded_Oacc(T_a,V_r_2)) )
     => c_member(T_a,V_xa_2,c_Wellfounded_Oacc(T_a,V_r_2)) ) )).

fof(fact_acc__downwards__aux,axiom,(
    ! [V_r_2,V_a_2,V_b_2,T_a] :
      ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_b_2,V_a_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2))
     => ( c_member(T_a,V_a_2,c_Wellfounded_Oacc(T_a,V_r_2))
       => c_member(T_a,V_b_2,c_Wellfounded_Oacc(T_a,V_r_2)) ) ) )).

fof(fact_acc__downwards,axiom,(
    ! [V_b_2,V_r_2,V_a_2,T_a] :
      ( c_member(T_a,V_a_2,c_Wellfounded_Oacc(T_a,V_r_2))
     => ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_b_2,V_a_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2))
       => c_member(T_a,V_b_2,c_Wellfounded_Oacc(T_a,V_r_2)) ) ) )).

fof(fact_irrefl__def,axiom,(
    ! [V_r_2,T_a] :
      ( c_Relation_Oirrefl(T_a,V_r_2)
    <=> ! [B_x] : ~ c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_x,B_x),V_r_2) ) )).

fof(fact_converse__in__Lin,axiom,(
    ! [V_La_2] :
      ( c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),c_Relation_Oconverse(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,V_La_2),c_Arrow__Order__Mirabelle_OLin)
    <=> c_member(tc_fun(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),tc_HOL_Obool),V_La_2,c_Arrow__Order__Mirabelle_OLin) ) )).

fof(fact_RangeI,axiom,(
    ! [V_r_2,V_b_2,V_a_2,T_b,T_a] :
      ( c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2),V_r_2)
     => c_member(T_b,V_b_2,c_Relation_ORange(T_a,T_b,V_r_2)) ) )).

fof(fact_Id__on__eqI,axiom,(
    ! [V_A_2,T_a,V_b_2,V_a_2] :
      ( V_a_2 = V_b_2
     => ( c_member(T_a,V_a_2,V_A_2)
       => c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Relation_OId__on(T_a,V_A_2)) ) ) )).

fof(fact_Id__on__iff,axiom,(
    ! [V_A_2,V_ya_2,V_xa_2,T_a] :
      ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_ya_2),c_Relation_OId__on(T_a,V_A_2))
    <=> ( V_xa_2 = V_ya_2
        & c_member(T_a,V_xa_2,V_A_2) ) ) )).

fof(fact_converse__Id__on,axiom,(
    ! [V_A_2,T_a] : c_Relation_Oconverse(T_a,T_a,c_Relation_OId__on(T_a,V_A_2)) = c_Relation_OId__on(T_a,V_A_2) )).

fof(fact_converse__converse,axiom,(
    ! [V_r_2,T_a,T_b] : c_Relation_Oconverse(T_b,T_a,c_Relation_Oconverse(T_a,T_b,V_r_2)) = V_r_2 )).

fof(fact_Range__Id__on,axiom,(
    ! [V_A_2,T_a] : c_Relation_ORange(T_a,T_a,c_Relation_OId__on(T_a,V_A_2)) = V_A_2 )).

fof(fact_converse__iff,axiom,(
    ! [V_r_2,V_b_2,V_a_2,T_b,T_a] :
      ( c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2),c_Relation_Oconverse(T_b,T_a,V_r_2))
    <=> c_member(tc_prod(T_b,T_a),c_Product__Type_OPair(T_b,T_a,V_b_2,V_a_2),V_r_2) ) )).

fof(fact_converseI,axiom,(
    ! [V_r_2,V_b_2,V_a_2,T_b,T_a] :
      ( c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2),V_r_2)
     => c_member(tc_prod(T_b,T_a),c_Product__Type_OPair(T_b,T_a,V_b_2,V_a_2),c_Relation_Oconverse(T_a,T_b,V_r_2)) ) )).

fof(fact_converseD,axiom,(
    ! [V_r_2,V_b_2,V_a_2,T_b,T_a] :
      ( c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2),c_Relation_Oconverse(T_b,T_a,V_r_2))
     => c_member(tc_prod(T_b,T_a),c_Product__Type_OPair(T_b,T_a,V_b_2,V_a_2),V_r_2) ) )).

fof(fact_Range__iff,axiom,(
    ! [V_r_2,T_b,V_a_2,T_a] :
      ( c_member(T_a,V_a_2,c_Relation_ORange(T_b,T_a,V_r_2))
    <=> ? [B_y] : c_member(tc_prod(T_b,T_a),c_Product__Type_OPair(T_b,T_a,B_y,V_a_2),V_r_2) ) )).

fof(fact_rtrancl_Ortrancl__refl,axiom,(
    ! [V_r_2,V_a_2,T_a] : c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_a_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)) )).

fof(fact_r__into__rtrancl,axiom,(
    ! [V_r_2,V_p_2,T_a] :
      ( c_member(tc_prod(T_a,T_a),V_p_2,V_r_2)
     => c_member(tc_prod(T_a,T_a),V_p_2,c_Transitive__Closure_Ortrancl(T_a,V_r_2)) ) )).

fof(fact_rtrancl__converseI,axiom,(
    ! [V_r_2,V_xa_2,V_ya_2,T_a] :
      ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_ya_2,V_xa_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2))
     => c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_ya_2),c_Transitive__Closure_Ortrancl(T_a,c_Relation_Oconverse(T_a,T_a,V_r_2))) ) )).

fof(fact_rtrancl__converseD,axiom,(
    ! [V_r_2,V_ya_2,V_xa_2,T_a] :
      ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_ya_2),c_Transitive__Closure_Ortrancl(T_a,c_Relation_Oconverse(T_a,T_a,V_r_2)))
     => c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_ya_2,V_xa_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)) ) )).

fof(fact_converse__rtrancl__into__rtrancl,axiom,(
    ! [V_c_2,V_r_2,V_b_2,V_a_2,T_a] :
      ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),V_r_2)
     => ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_b_2,V_c_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2))
       => c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_c_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)) ) ) )).

fof(fact_rtrancl_Ortrancl__into__rtrancl,axiom,(
    ! [V_c_2,V_r_2,V_b_2,V_a_2,T_a] :
      ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2))
     => ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_b_2,V_c_2),V_r_2)
       => c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_c_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)) ) ) )).

fof(fact_rtrancl__trans,axiom,(
    ! [V_c_2,V_r_2,V_b_2,V_a_2,T_a] :
      ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2))
     => ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_b_2,V_c_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2))
       => c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_c_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)) ) ) )).

fof(fact_rtrancl__converse,axiom,(
    ! [V_r_2,T_a] : c_Transitive__Closure_Ortrancl(T_a,c_Relation_Oconverse(T_a,T_a,V_r_2)) = c_Relation_Oconverse(T_a,T_a,c_Transitive__Closure_Ortrancl(T_a,V_r_2)) )).

fof(fact_Id__onE,axiom,(
    ! [V_A_2,V_c_2,T_a] :
      ( c_member(tc_prod(T_a,T_a),V_c_2,c_Relation_OId__on(T_a,V_A_2))
     => ~ ! [B_x] :
            ( c_member(T_a,B_x,V_A_2)
           => V_c_2 != c_Product__Type_OPair(T_a,T_a,B_x,B_x) ) ) )).

fof(fact_rtrancl__idemp,axiom,(
    ! [V_r_2,T_a] : c_Transitive__Closure_Ortrancl(T_a,c_Transitive__Closure_Ortrancl(T_a,V_r_2)) = c_Transitive__Closure_Ortrancl(T_a,V_r_2) )).

fof(fact_RangeE,axiom,(
    ! [V_r_2,T_b,V_b_2,T_a] :
      ( c_member(T_a,V_b_2,c_Relation_ORange(T_b,T_a,V_r_2))
     => ~ ! [B_x] : ~ c_member(tc_prod(T_b,T_a),c_Product__Type_OPair(T_b,T_a,B_x,V_b_2),V_r_2) ) )).

fof(fact_single__valued__confluent,axiom,(
    ! [V_z_2,V_ya_2,V_xa_2,V_r_2,T_a] :
      ( c_Relation_Osingle__valued(T_a,T_a,V_r_2)
     => ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_ya_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2))
       => ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_z_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2))
         => ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_ya_2,V_z_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2))
            | c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_z_2,V_ya_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)) ) ) ) ) )).

fof(fact_converseE,axiom,(
    ! [V_r_2,V_yx_2,T_b,T_a] :
      ( c_member(tc_prod(T_a,T_b),V_yx_2,c_Relation_Oconverse(T_b,T_a,V_r_2))
     => ~ ! [B_x,B_y] :
            ( V_yx_2 = c_Product__Type_OPair(T_a,T_b,B_y,B_x)
           => ~ c_member(tc_prod(T_b,T_a),c_Product__Type_OPair(T_b,T_a,B_x,B_y),V_r_2) ) ) )).

fof(fact_Not__Domain__rtrancl,axiom,(
    ! [V_ya_2,V_R_2,V_xa_2,T_a] :
      ( ~ c_member(T_a,V_xa_2,c_Relation_ODomain(T_a,T_a,V_R_2))
     => ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_ya_2),c_Transitive__Closure_Ortrancl(T_a,V_R_2))
      <=> V_xa_2 = V_ya_2 ) ) )).

fof(fact_DomainI,axiom,(
    ! [V_r_2,V_b_2,V_a_2,T_b,T_a] :
      ( c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2),V_r_2)
     => c_member(T_a,V_a_2,c_Relation_ODomain(T_a,T_b,V_r_2)) ) )).

fof(fact_rtrancl__induct2,axiom,(
    ! [V_P_2,V_r_2,V_by_2,V_bx_2,V_ay_2,V_ax_2,T_b,T_a] :
      ( c_member(tc_prod(tc_prod(T_a,T_b),tc_prod(T_a,T_b)),c_Product__Type_OPair(tc_prod(T_a,T_b),tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_ax_2,V_ay_2),c_Product__Type_OPair(T_a,T_b,V_bx_2,V_by_2)),c_Transitive__Closure_Ortrancl(tc_prod(T_a,T_b),V_r_2))
     => ( hBOOL(hAPP(hAPP(V_P_2,V_ax_2),V_ay_2))
       => ( ! [B_a,B_b] :
              ( c_member(tc_prod(tc_prod(T_a,T_b),tc_prod(T_a,T_b)),c_Product__Type_OPair(tc_prod(T_a,T_b),tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_ax_2,V_ay_2),c_Product__Type_OPair(T_a,T_b,B_a,B_b)),c_Transitive__Closure_Ortrancl(tc_prod(T_a,T_b),V_r_2))
             => ! [B_aa,B_ba] :
                  ( c_member(tc_prod(tc_prod(T_a,T_b),tc_prod(T_a,T_b)),c_Product__Type_OPair(tc_prod(T_a,T_b),tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,B_a,B_b),c_Product__Type_OPair(T_a,T_b,B_aa,B_ba)),V_r_2)
                 => ( hBOOL(hAPP(hAPP(V_P_2,B_a),B_b))
                   => hBOOL(hAPP(hAPP(V_P_2,B_aa),B_ba)) ) ) )
         => hBOOL(hAPP(hAPP(V_P_2,V_bx_2),V_by_2)) ) ) ) )).

fof(fact_converse__rtrancl__induct2,axiom,(
    ! [V_P_2,V_r_2,V_by_2,V_bx_2,V_ay_2,V_ax_2,T_b,T_a] :
      ( c_member(tc_prod(tc_prod(T_a,T_b),tc_prod(T_a,T_b)),c_Product__Type_OPair(tc_prod(T_a,T_b),tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_ax_2,V_ay_2),c_Product__Type_OPair(T_a,T_b,V_bx_2,V_by_2)),c_Transitive__Closure_Ortrancl(tc_prod(T_a,T_b),V_r_2))
     => ( hBOOL(hAPP(hAPP(V_P_2,V_bx_2),V_by_2))
       => ( ! [B_a,B_b,B_aa,B_ba] :
              ( c_member(tc_prod(tc_prod(T_a,T_b),tc_prod(T_a,T_b)),c_Product__Type_OPair(tc_prod(T_a,T_b),tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,B_a,B_b),c_Product__Type_OPair(T_a,T_b,B_aa,B_ba)),V_r_2)
             => ( c_member(tc_prod(tc_prod(T_a,T_b),tc_prod(T_a,T_b)),c_Product__Type_OPair(tc_prod(T_a,T_b),tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,B_aa,B_ba),c_Product__Type_OPair(T_a,T_b,V_bx_2,V_by_2)),c_Transitive__Closure_Ortrancl(tc_prod(T_a,T_b),V_r_2))
               => ( hBOOL(hAPP(hAPP(V_P_2,B_aa),B_ba))
                 => hBOOL(hAPP(hAPP(V_P_2,B_a),B_b)) ) ) )
         => hBOOL(hAPP(hAPP(V_P_2,V_ax_2),V_ay_2)) ) ) ) )).

fof(fact_converse__rtranclE2,axiom,(
    ! [V_r_2,V_zb_2,V_za_2,V_xb_2,V_xa_2,T_b,T_a] :
      ( c_member(tc_prod(tc_prod(T_a,T_b),tc_prod(T_a,T_b)),c_Product__Type_OPair(tc_prod(T_a,T_b),tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_xa_2,V_xb_2),c_Product__Type_OPair(T_a,T_b,V_za_2,V_zb_2)),c_Transitive__Closure_Ortrancl(tc_prod(T_a,T_b),V_r_2))
     => ( c_Product__Type_OPair(T_a,T_b,V_xa_2,V_xb_2) != c_Product__Type_OPair(T_a,T_b,V_za_2,V_zb_2)
       => ~ ! [B_a,B_b] :
              ( c_member(tc_prod(tc_prod(T_a,T_b),tc_prod(T_a,T_b)),c_Product__Type_OPair(tc_prod(T_a,T_b),tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_xa_2,V_xb_2),c_Product__Type_OPair(T_a,T_b,B_a,B_b)),V_r_2)
             => ~ c_member(tc_prod(tc_prod(T_a,T_b),tc_prod(T_a,T_b)),c_Product__Type_OPair(tc_prod(T_a,T_b),tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,B_a,B_b),c_Product__Type_OPair(T_a,T_b,V_za_2,V_zb_2)),c_Transitive__Closure_Ortrancl(tc_prod(T_a,T_b),V_r_2)) ) ) ) )).

fof(fact_rtranclE,axiom,(
    ! [V_r_2,V_b_2,V_a_2,T_a] :
      ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2))
     => ( V_a_2 != V_b_2
       => ~ ! [B_y] :
              ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,B_y),c_Transitive__Closure_Ortrancl(T_a,V_r_2))
             => ~ c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_y,V_b_2),V_r_2) ) ) ) )).

fof(fact_Domain__Id__on,axiom,(
    ! [V_A_2,T_a] : c_Relation_ODomain(T_a,T_a,c_Relation_OId__on(T_a,V_A_2)) = V_A_2 )).

fof(fact_single__valued__Id__on,axiom,(
    ! [V_A_2,T_a] : c_Relation_Osingle__valued(T_a,T_a,c_Relation_OId__on(T_a,V_A_2)) )).

fof(fact_Range__def,axiom,(
    ! [V_r_2,T_a,T_b] : c_Relation_ORange(T_b,T_a,V_r_2) = c_Relation_ODomain(T_a,T_b,c_Relation_Oconverse(T_b,T_a,V_r_2)) )).

fof(fact_Domain__converse,axiom,(
    ! [V_r_2,T_b,T_a] : c_Relation_ODomain(T_a,T_b,c_Relation_Oconverse(T_b,T_a,V_r_2)) = c_Relation_ORange(T_b,T_a,V_r_2) )).

fof(fact_Range__converse,axiom,(
    ! [V_r_2,T_a,T_b] : c_Relation_ORange(T_b,T_a,c_Relation_Oconverse(T_a,T_b,V_r_2)) = c_Relation_ODomain(T_a,T_b,V_r_2) )).

fof(fact_single__valued__def,axiom,(
    ! [V_r_2,T_b,T_a] :
      ( c_Relation_Osingle__valued(T_a,T_b,V_r_2)
    <=> ! [B_x,B_y] :
          ( c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,B_x,B_y),V_r_2)
         => ! [B_z] :
              ( c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,B_x,B_z),V_r_2)
             => B_y = B_z ) ) ) )).

fof(fact_single__valuedD,axiom,(
    ! [V_z_2,V_ya_2,V_xa_2,V_r_2,T_b,T_a] :
      ( c_Relation_Osingle__valued(T_a,T_b,V_r_2)
     => ( c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_xa_2,V_ya_2),V_r_2)
       => ( c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_xa_2,V_z_2),V_r_2)
         => V_ya_2 = V_z_2 ) ) ) )).

fof(fact_Domain__iff,axiom,(
    ! [V_r_2,T_b,V_a_2,T_a] :
      ( c_member(T_a,V_a_2,c_Relation_ODomain(T_a,T_b,V_r_2))
    <=> ? [B_y] : c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_a_2,B_y),V_r_2) ) )).

fof(fact_DomainE,axiom,(
    ! [V_r_2,T_b,V_a_2,T_a] :
      ( c_member(T_a,V_a_2,c_Relation_ODomain(T_a,T_b,V_r_2))
     => ~ ! [B_y] : ~ c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_a_2,B_y),V_r_2) ) )).

fof(fact_single__valuedI,axiom,(
    ! [V_r_2,T_b,T_a] :
      ( ! [B_x,B_y] :
          ( c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,B_x,B_y),V_r_2)
         => ! [B_z] :
              ( c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,B_x,B_z),V_r_2)
             => B_y = B_z ) )
     => c_Relation_Osingle__valued(T_a,T_b,V_r_2) ) )).

fof(fact_rtrancl__induct,axiom,(
    ! [V_P_2,V_r_2,V_b_2,V_a_2,T_a] :
      ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2))
     => ( hBOOL(hAPP(V_P_2,V_a_2))
       => ( ! [B_y] :
              ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,B_y),c_Transitive__Closure_Ortrancl(T_a,V_r_2))
             => ! [B_z] :
                  ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_y,B_z),V_r_2)
                 => ( hBOOL(hAPP(V_P_2,B_y))
                   => hBOOL(hAPP(V_P_2,B_z)) ) ) )
         => hBOOL(hAPP(V_P_2,V_b_2)) ) ) ) )).

fof(fact_converse__rtrancl__induct,axiom,(
    ! [V_P_2,V_r_2,V_b_2,V_a_2,T_a] :
      ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2))
     => ( hBOOL(hAPP(V_P_2,V_b_2))
       => ( ! [B_y,B_z] :
              ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_y,B_z),V_r_2)
             => ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_z,V_b_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2))
               => ( hBOOL(hAPP(V_P_2,B_z))
                 => hBOOL(hAPP(V_P_2,B_y)) ) ) )
         => hBOOL(hAPP(V_P_2,V_a_2)) ) ) ) )).

fof(fact_converse__rtranclE,axiom,(
    ! [V_r_2,V_z_2,V_xa_2,T_a] :
      ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_z_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2))
     => ( V_xa_2 != V_z_2
       => ~ ! [B_y] :
              ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,B_y),V_r_2)
             => ~ c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_y,V_z_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)) ) ) ) )).

fof(fact_total__on__def,axiom,(
    ! [V_r_2,V_A_2,T_a] :
      ( c_Relation_Ototal__on(T_a,V_A_2,V_r_2)
    <=> ! [B_x] :
          ( c_member(T_a,B_x,V_A_2)
         => ! [B_xa] :
              ( c_member(T_a,B_xa,V_A_2)
             => ( B_x != B_xa
               => ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_x,B_xa),V_r_2)
                  | c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_xa,B_x),V_r_2) ) ) ) ) ) )).

fof(fact_in__inv__image,axiom,(
    ! [V_f_2,V_r_2,T_b,V_ya_2,V_xa_2,T_a] :
      ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_ya_2),hAPP(c_Relation_Oinv__image(T_b,T_a,V_r_2),V_f_2))
    <=> c_member(tc_prod(T_b,T_b),c_Product__Type_OPair(T_b,T_b,hAPP(V_f_2,V_xa_2),hAPP(V_f_2,V_ya_2)),V_r_2) ) )).

fof(fact_refl__onD,axiom,(
    ! [V_a_2,V_r_2,V_A_2,T_a] :
      ( c_Relation_Orefl__on(T_a,V_A_2,V_r_2)
     => ( c_member(T_a,V_a_2,V_A_2)
       => c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_a_2),V_r_2) ) ) )).

fof(fact_refl__on__converse,axiom,(
    ! [V_r_2,V_A_2,T_a] :
      ( c_Relation_Orefl__on(T_a,V_A_2,c_Relation_Oconverse(T_a,T_a,V_r_2))
    <=> c_Relation_Orefl__on(T_a,V_A_2,V_r_2) ) )).

fof(fact_refl__on__Id__on,axiom,(
    ! [V_A_2,T_a] : c_Relation_Orefl__on(T_a,V_A_2,c_Relation_OId__on(T_a,V_A_2)) )).

fof(fact_converse__inv__image,axiom,(
    ! [V_f_2,V_R_2,T_b,T_a] : c_Relation_Oconverse(T_a,T_a,hAPP(c_Relation_Oinv__image(T_b,T_a,V_R_2),V_f_2)) = hAPP(c_Relation_Oinv__image(T_b,T_a,c_Relation_Oconverse(T_b,T_b,V_R_2)),V_f_2) )).

fof(fact_total__on__converse,axiom,(
    ! [V_r_2,V_A_2,T_a] :
      ( c_Relation_Ototal__on(T_a,V_A_2,c_Relation_Oconverse(T_a,T_a,V_r_2))
    <=> c_Relation_Ototal__on(T_a,V_A_2,V_r_2) ) )).

fof(fact_refl__onD2,axiom,(
    ! [V_ya_2,V_xa_2,V_r_2,V_A_2,T_a] :
      ( c_Relation_Orefl__on(T_a,V_A_2,V_r_2)
     => ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_ya_2),V_r_2)
       => c_member(T_a,V_ya_2,V_A_2) ) ) )).

fof(fact_refl__onD1,axiom,(
    ! [V_ya_2,V_xa_2,V_r_2,V_A_2,T_a] :
      ( c_Relation_Orefl__on(T_a,V_A_2,V_r_2)
     => ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_ya_2),V_r_2)
       => c_member(T_a,V_xa_2,V_A_2) ) ) )).

fof(fact_trancl__into__rtrancl,axiom,(
    ! [V_r_2,V_b_2,V_a_2,T_a] :
      ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Otrancl(T_a,V_r_2))
     => c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)) ) )).

fof(fact_IdI,axiom,(
    ! [V_a_2,T_a] : c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_a_2),c_Relation_OId(T_a)) )).

fof(fact_antisymD,axiom,(
    ! [V_b_2,V_a_2,V_r_2,T_a] :
      ( c_Relation_Oantisym(T_a,V_r_2)
     => ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),V_r_2)
       => ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_b_2,V_a_2),V_r_2)
         => V_a_2 = V_b_2 ) ) ) )).

fof(fact_antisym__def,axiom,(
    ! [V_r_2,T_a] :
      ( c_Relation_Oantisym(T_a,V_r_2)
    <=> ! [B_x,B_y] :
          ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_x,B_y),V_r_2)
         => ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_y,B_x),V_r_2)
           => B_x = B_y ) ) ) )).

fof(fact_trancl__converseD,axiom,(
    ! [V_r_2,V_ya_2,V_xa_2,T_a] :
      ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_ya_2),c_Transitive__Closure_Otrancl(T_a,c_Relation_Oconverse(T_a,T_a,V_r_2)))
     => c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_ya_2),c_Relation_Oconverse(T_a,T_a,c_Transitive__Closure_Otrancl(T_a,V_r_2))) ) )).

fof(fact_trancl__converseI,axiom,(
    ! [V_r_2,V_ya_2,V_xa_2,T_a] :
      ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_ya_2),c_Relation_Oconverse(T_a,T_a,c_Transitive__Closure_Otrancl(T_a,V_r_2)))
     => c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_ya_2),c_Transitive__Closure_Otrancl(T_a,c_Relation_Oconverse(T_a,T_a,V_r_2))) ) )).

fof(fact_rtrancl__trancl__trancl,axiom,(
    ! [V_z_2,V_r_2,V_ya_2,V_xa_2,T_a] :
      ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_ya_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2))
     => ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_ya_2,V_z_2),c_Transitive__Closure_Otrancl(T_a,V_r_2))
       => c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_z_2),c_Transitive__Closure_Otrancl(T_a,V_r_2)) ) ) )).

fof(fact_trancl__rtrancl__trancl,axiom,(
    ! [V_c_2,V_r_2,V_b_2,V_a_2,T_a] :
      ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Otrancl(T_a,V_r_2))
     => ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_b_2,V_c_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2))
       => c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_c_2),c_Transitive__Closure_Otrancl(T_a,V_r_2)) ) ) )).

fof(fact_trancl_Or__into__trancl,axiom,(
    ! [V_r_2,V_b_2,V_a_2,T_a] :
      ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),V_r_2)
     => c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Otrancl(T_a,V_r_2)) ) )).

fof(fact_antisym__Id,axiom,(
    ! [T_a] : c_Relation_Oantisym(T_a,c_Relation_OId(T_a)) )).

fof(fact_r__into__trancl_H,axiom,(
    ! [V_r_2,V_p_2,T_a] :
      ( c_member(tc_prod(T_a,T_a),V_p_2,V_r_2)
     => c_member(tc_prod(T_a,T_a),V_p_2,c_Transitive__Closure_Otrancl(T_a,V_r_2)) ) )).

fof(fact_trancl__rtrancl__absorb,axiom,(
    ! [V_R_2,T_a] : c_Transitive__Closure_Ortrancl(T_a,c_Transitive__Closure_Otrancl(T_a,V_R_2)) = c_Transitive__Closure_Ortrancl(T_a,V_R_2) )).

fof(fact_rtrancl__trancl__absorb,axiom,(
    ! [V_R_2,T_a] : c_Transitive__Closure_Otrancl(T_a,c_Transitive__Closure_Ortrancl(T_a,V_R_2)) = c_Transitive__Closure_Ortrancl(T_a,V_R_2) )).

fof(fact_trancl__converse,axiom,(
    ! [V_r_2,T_a] : c_Transitive__Closure_Otrancl(T_a,c_Relation_Oconverse(T_a,T_a,V_r_2)) = c_Relation_Oconverse(T_a,T_a,c_Transitive__Closure_Otrancl(T_a,V_r_2)) )).

fof(fact_trancl__domain,axiom,(
    ! [V_r_2,T_a] : c_Relation_ODomain(T_a,T_a,c_Transitive__Closure_Otrancl(T_a,V_r_2)) = c_Relation_ODomain(T_a,T_a,V_r_2) )).

fof(fact_trancl__range,axiom,(
    ! [V_r_2,T_a] : c_Relation_ORange(T_a,T_a,c_Transitive__Closure_Otrancl(T_a,V_r_2)) = c_Relation_ORange(T_a,T_a,V_r_2) )).

fof(fact_converse__Id,axiom,(
    ! [T_a] : c_Relation_Oconverse(T_a,T_a,c_Relation_OId(T_a)) = c_Relation_OId(T_a) )).

fof(fact_single__valued__Id,axiom,(
    ! [T_a] : c_Relation_Osingle__valued(T_a,T_a,c_Relation_OId(T_a)) )).

fof(fact_antisym__converse,axiom,(
    ! [V_r_2,T_a] :
      ( c_Relation_Oantisym(T_a,c_Relation_Oconverse(T_a,T_a,V_r_2))
    <=> c_Relation_Oantisym(T_a,V_r_2) ) )).

fof(fact_r__r__into__trancl,axiom,(
    ! [V_c_2,V_R_2,V_b_2,V_a_2,T_a] :
      ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),V_R_2)
     => ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_b_2,V_c_2),V_R_2)
       => c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_c_2),c_Transitive__Closure_Otrancl(T_a,V_R_2)) ) ) )).

fof(fact_trancl__into__trancl2,axiom,(
    ! [V_c_2,V_r_2,V_b_2,V_a_2,T_a] :
      ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),V_r_2)
     => ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_b_2,V_c_2),c_Transitive__Closure_Otrancl(T_a,V_r_2))
       => c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_c_2),c_Transitive__Closure_Otrancl(T_a,V_r_2)) ) ) )).

fof(fact_Transitive__Closure_Otrancl__into__trancl,axiom,(
    ! [V_c_2,V_r_2,V_b_2,V_a_2,T_a] :
      ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Otrancl(T_a,V_r_2))
     => ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_b_2,V_c_2),V_r_2)
       => c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_c_2),c_Transitive__Closure_Otrancl(T_a,V_r_2)) ) ) )).

fof(fact_trancl__trans,axiom,(
    ! [V_c_2,V_r_2,V_b_2,V_a_2,T_a] :
      ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Otrancl(T_a,V_r_2))
     => ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_b_2,V_c_2),c_Transitive__Closure_Otrancl(T_a,V_r_2))
       => c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_c_2),c_Transitive__Closure_Otrancl(T_a,V_r_2)) ) ) )).

fof(fact_antisym__Id__on,axiom,(
    ! [V_A_2,T_a] : c_Relation_Oantisym(T_a,c_Relation_OId__on(T_a,V_A_2)) )).

fof(fact_pair__in__Id__conv,axiom,(
    ! [V_b_2,V_a_2,T_a] :
      ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Relation_OId(T_a))
    <=> V_a_2 = V_b_2 ) )).

fof(fact_rtrancl__eq__or__trancl,axiom,(
    ! [V_R_2,V_ya_2,V_xa_2,T_a] :
      ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_ya_2),c_Transitive__Closure_Ortrancl(T_a,V_R_2))
    <=> ( V_xa_2 = V_ya_2
        | ( V_xa_2 != V_ya_2
          & c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_ya_2),c_Transitive__Closure_Otrancl(T_a,V_R_2)) ) ) ) )).

fof(fact_rtrancl__into__trancl2,axiom,(
    ! [V_c_2,V_r_2,V_b_2,V_a_2,T_a] :
      ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),V_r_2)
     => ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_b_2,V_c_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2))
       => c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_c_2),c_Transitive__Closure_Otrancl(T_a,V_r_2)) ) ) )).

fof(fact_rtranclD,axiom,(
    ! [V_R_2,V_b_2,V_a_2,T_a] :
      ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Ortrancl(T_a,V_R_2))
     => ( V_a_2 = V_b_2
        | ( V_a_2 != V_b_2
          & c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Otrancl(T_a,V_R_2)) ) ) ) )).

fof(fact_rtrancl__into__trancl1,axiom,(
    ! [V_c_2,V_r_2,V_b_2,V_a_2,T_a] :
      ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2))
     => ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_b_2,V_c_2),V_r_2)
       => c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_c_2),c_Transitive__Closure_Otrancl(T_a,V_r_2)) ) ) )).

fof(fact_tranclD2,axiom,(
    ! [V_R_2,V_ya_2,V_xa_2,T_a] :
      ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_ya_2),c_Transitive__Closure_Otrancl(T_a,V_R_2))
     => ? [B_z] :
          ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,B_z),c_Transitive__Closure_Ortrancl(T_a,V_R_2))
          & c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_z,V_ya_2),V_R_2) ) ) )).

fof(fact_tranclD,axiom,(
    ! [V_R_2,V_ya_2,V_xa_2,T_a] :
      ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_ya_2),c_Transitive__Closure_Otrancl(T_a,V_R_2))
     => ? [B_z] :
          ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,B_z),V_R_2)
          & c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_z,V_ya_2),c_Transitive__Closure_Ortrancl(T_a,V_R_2)) ) ) )).

fof(fact_IdE,axiom,(
    ! [V_p_2,T_a] :
      ( c_member(tc_prod(T_a,T_a),V_p_2,c_Relation_OId(T_a))
     => ~ ! [B_x] : V_p_2 != c_Product__Type_OPair(T_a,T_a,B_x,B_x) ) )).

fof(fact_acyclic__def,axiom,(
    ! [V_r_2,T_a] :
      ( c_Wellfounded_Oacyclic(T_a,V_r_2)
    <=> ! [B_x] : ~ c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_x,B_x),c_Transitive__Closure_Otrancl(T_a,V_r_2)) ) )).

fof(fact_antisymI,axiom,(
    ! [V_r_2,T_a] :
      ( ! [B_x,B_y] :
          ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_x,B_y),V_r_2)
         => ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_y,B_x),V_r_2)
           => B_x = B_y ) )
     => c_Relation_Oantisym(T_a,V_r_2) ) )).

fof(fact_irrefl__trancl__rD,axiom,(
    ! [V_ya_2,V_xa_2,V_r_2,T_a] :
      ( ! [B_x] : ~ c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_x,B_x),c_Transitive__Closure_Otrancl(T_a,V_r_2))
     => ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_ya_2),V_r_2)
       => V_xa_2 != V_ya_2 ) ) )).

fof(fact_converse__tranclE,axiom,(
    ! [V_r_2,V_z_2,V_xa_2,T_a] :
      ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_z_2),c_Transitive__Closure_Otrancl(T_a,V_r_2))
     => ( ~ c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_z_2),V_r_2)
       => ~ ! [B_y] :
              ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,B_y),V_r_2)
             => ~ c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_y,V_z_2),c_Transitive__Closure_Otrancl(T_a,V_r_2)) ) ) ) )).

fof(fact_acyclic__converse,axiom,(
    ! [V_r_2,T_a] :
      ( c_Wellfounded_Oacyclic(T_a,c_Relation_Oconverse(T_a,T_a,V_r_2))
    <=> c_Wellfounded_Oacyclic(T_a,V_r_2) ) )).

fof(fact_acyclic__impl__antisym__rtrancl,axiom,(
    ! [V_r_2,T_a] :
      ( c_Wellfounded_Oacyclic(T_a,V_r_2)
     => c_Relation_Oantisym(T_a,c_Transitive__Closure_Ortrancl(T_a,V_r_2)) ) )).

fof(fact_acyclicI,axiom,(
    ! [V_r_2,T_a] :
      ( ! [B_x] : ~ c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_x,B_x),c_Transitive__Closure_Otrancl(T_a,V_r_2))
     => c_Wellfounded_Oacyclic(T_a,V_r_2) ) )).

fof(fact_acyclic__insert,axiom,(
    ! [V_r_2,V_xa_2,V_ya_2,T_a] :
      ( c_Wellfounded_Oacyclic(T_a,c_Set_Oinsert(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_ya_2,V_xa_2),V_r_2))
    <=> ( c_Wellfounded_Oacyclic(T_a,V_r_2)
        & ~ c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_ya_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)) ) ) )).

fof(fact_tranclE,axiom,(
    ! [V_r_2,V_b_2,V_a_2,T_a] :
      ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Otrancl(T_a,V_r_2))
     => ( ~ c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),V_r_2)
       => ~ ! [B_c] :
              ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,B_c),c_Transitive__Closure_Otrancl(T_a,V_r_2))
             => ~ c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_c,V_b_2),V_r_2) ) ) ) )).

fof(fact_reflcl__trancl,axiom,(
    ! [V_r_2,T_a] : c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),c_Transitive__Closure_Otrancl(T_a,V_r_2),c_Relation_OId(T_a)) = c_Transitive__Closure_Ortrancl(T_a,V_r_2) )).

fof(fact_trancl__reflcl,axiom,(
    ! [V_r_2,T_a] : c_Transitive__Closure_Otrancl(T_a,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,c_Relation_OId(T_a))) = c_Transitive__Closure_Ortrancl(T_a,V_r_2) )).

fof(fact_irrefl__diff__Id,axiom,(
    ! [V_r_2,T_a] : c_Relation_Oirrefl(T_a,c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,c_Relation_OId(T_a))) )).

fof(fact_sup1E,axiom,(
    ! [V_xa_2,V_B_2,V_A_2,T_a] :
      ( hBOOL(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_xa_2))
     => ( ~ hBOOL(hAPP(V_A_2,V_xa_2))
       => hBOOL(hAPP(V_B_2,V_xa_2)) ) ) )).

fof(fact_sup1CI,axiom,(
    ! [T_a,V_A_2,V_xa_2,V_B_2] :
      ( ( ~ hBOOL(hAPP(V_B_2,V_xa_2))
       => hBOOL(hAPP(V_A_2,V_xa_2)) )
     => hBOOL(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_xa_2)) ) )).

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

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

fof(fact_UnE,axiom,(
    ! [V_B_2,V_A_2,V_c_2,T_a] :
      ( c_member(T_a,V_c_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2))
     => ( ~ c_member(T_a,V_c_2,V_A_2)
       => c_member(T_a,V_c_2,V_B_2) ) ) )).

fof(fact_UnCI,axiom,(
    ! [V_A_2,V_B_2,V_c_2,T_a] :
      ( ( ~ c_member(T_a,V_c_2,V_B_2)
       => c_member(T_a,V_c_2,V_A_2) )
     => c_member(T_a,V_c_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)) ) )).

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

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

fof(fact_refl__on__Un,axiom,(
    ! [V_s_2,V_B_2,V_r_2,V_A_2,T_a] :
      ( c_Relation_Orefl__on(T_a,V_A_2,V_r_2)
     => ( c_Relation_Orefl__on(T_a,V_B_2,V_s_2)
       => c_Relation_Orefl__on(T_a,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,V_s_2)) ) ) )).

fof(fact_sup1I1,axiom,(
    ! [V_B_2,T_a,V_xa_2,V_A_2] :
      ( hBOOL(hAPP(V_A_2,V_xa_2))
     => hBOOL(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_xa_2)) ) )).

fof(fact_sup1I2,axiom,(
    ! [V_A_2,T_a,V_xa_2,V_B_2] :
      ( hBOOL(hAPP(V_B_2,V_xa_2))
     => hBOOL(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_xa_2)) ) )).

fof(fact_Un__absorb,axiom,(
    ! [V_A_2,T_a] : c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_A_2) = V_A_2 )).

fof(fact_Un__commute,axiom,(
    ! [V_B_2,V_A_2,T_a] : c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_A_2) )).

fof(fact_insert__absorb2,axiom,(
    ! [V_A_2,V_xa_2,T_a] : c_Set_Oinsert(T_a,V_xa_2,c_Set_Oinsert(T_a,V_xa_2,V_A_2)) = c_Set_Oinsert(T_a,V_xa_2,V_A_2) )).

fof(fact_Un__Diff__cancel,axiom,(
    ! [V_B_2,V_A_2,T_a] : c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_A_2)) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) )).

fof(fact_Un__left__absorb,axiom,(
    ! [V_B_2,V_A_2,T_a] : c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) )).

fof(fact_insert__commute,axiom,(
    ! [V_A_2,V_ya_2,V_xa_2,T_a] : c_Set_Oinsert(T_a,V_xa_2,c_Set_Oinsert(T_a,V_ya_2,V_A_2)) = c_Set_Oinsert(T_a,V_ya_2,c_Set_Oinsert(T_a,V_xa_2,V_A_2)) )).

fof(fact_Un__insert__right,axiom,(
    ! [V_B_2,V_a_2,V_A_2,T_a] : c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Set_Oinsert(T_a,V_a_2,V_B_2)) = c_Set_Oinsert(T_a,V_a_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,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_a] : c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2)) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_B_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_C_2)) )).

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

fof(fact_Un__Diff__cancel2,axiom,(
    ! [V_A_2,V_B_2,T_a] : c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_A_2),V_A_2) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_A_2) )).

fof(fact_Un__insert__left,axiom,(
    ! [V_C_2,V_B_2,V_a_2,T_a] : c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),c_Set_Oinsert(T_a,V_a_2,V_B_2),V_C_2) = c_Set_Oinsert(T_a,V_a_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2)) )).

fof(fact_Un__assoc,axiom,(
    ! [V_C_2,V_B_2,V_A_2,T_a] : c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_C_2) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2)) )).

fof(fact_Un__Diff,axiom,(
    ! [V_C_2,V_B_2,V_A_2,T_a] : c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_C_2) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_C_2),c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2)) )).

fof(fact_bex__Un,axiom,(
    ! [V_P_2,V_B_2,V_A_2,T_a] :
      ( ? [B_x] :
          ( c_member(T_a,B_x,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2))
          & hBOOL(hAPP(V_P_2,B_x)) )
    <=> ( ? [B_x] :
            ( c_member(T_a,B_x,V_A_2)
            & hBOOL(hAPP(V_P_2,B_x)) )
        | ? [B_x] :
            ( c_member(T_a,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_a] :
      ( ! [B_x] :
          ( c_member(T_a,B_x,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2))
         => hBOOL(hAPP(V_P_2,B_x)) )
    <=> ( ! [B_x] :
            ( c_member(T_a,B_x,V_A_2)
           => hBOOL(hAPP(V_P_2,B_x)) )
        & ! [B_x] :
            ( c_member(T_a,B_x,V_B_2)
           => hBOOL(hAPP(V_P_2,B_x)) ) ) ) )).

fof(fact_insert__code,axiom,(
    ! [V_xa_2,V_A_2,V_ya_2,T_a] :
      ( hBOOL(hAPP(c_Set_Oinsert(T_a,V_ya_2,V_A_2),V_xa_2))
    <=> ( V_ya_2 = V_xa_2
        | hBOOL(hAPP(V_A_2,V_xa_2)) ) ) )).

fof(fact_insert__Diff__if,axiom,(
    ! [V_A_2,V_B_2,V_xa_2,T_a] :
      ( ( c_member(T_a,V_xa_2,V_B_2)
       => c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),c_Set_Oinsert(T_a,V_xa_2,V_A_2),V_B_2) = c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) )
      & ( ~ c_member(T_a,V_xa_2,V_B_2)
       => c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),c_Set_Oinsert(T_a,V_xa_2,V_A_2),V_B_2) = c_Set_Oinsert(T_a,V_xa_2,c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)) ) ) )).

fof(fact_insert__Diff1,axiom,(
    ! [V_A_2,V_B_2,V_xa_2,T_a] :
      ( c_member(T_a,V_xa_2,V_B_2)
     => c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),c_Set_Oinsert(T_a,V_xa_2,V_A_2),V_B_2) = c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) ) )).

fof(fact_Range__Un__eq,axiom,(
    ! [V_B_2,V_A_2,T_a,T_b] : c_Relation_ORange(T_b,T_a,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_a),tc_HOL_Obool),V_A_2,V_B_2)) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),c_Relation_ORange(T_b,T_a,V_A_2),c_Relation_ORange(T_b,T_a,V_B_2)) )).

fof(fact_Domain__Un__eq,axiom,(
    ! [V_B_2,V_A_2,T_b,T_a] : c_Relation_ODomain(T_a,T_b,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),V_A_2,V_B_2)) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),c_Relation_ODomain(T_a,T_b,V_A_2),c_Relation_ODomain(T_a,T_b,V_B_2)) )).

fof(fact_Domain__insert,axiom,(
    ! [V_r_2,V_b_2,V_a_2,T_b,T_a] : c_Relation_ODomain(T_a,T_b,c_Set_Oinsert(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2),V_r_2)) = c_Set_Oinsert(T_a,V_a_2,c_Relation_ODomain(T_a,T_b,V_r_2)) )).

fof(fact_Range__insert,axiom,(
    ! [V_r_2,V_b_2,V_a_2,T_a,T_b] : c_Relation_ORange(T_b,T_a,c_Set_Oinsert(tc_prod(T_b,T_a),c_Product__Type_OPair(T_b,T_a,V_a_2,V_b_2),V_r_2)) = c_Set_Oinsert(T_a,V_b_2,c_Relation_ORange(T_b,T_a,V_r_2)) )).

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

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

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

fof(fact_Un__iff,axiom,(
    ! [V_B_2,V_A_2,V_c_2,T_a] :
      ( c_member(T_a,V_c_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2))
    <=> ( c_member(T_a,V_c_2,V_A_2)
        | c_member(T_a,V_c_2,V_B_2) ) ) )).

fof(fact_UnI1,axiom,(
    ! [V_B_2,V_A_2,V_c_2,T_a] :
      ( c_member(T_a,V_c_2,V_A_2)
     => c_member(T_a,V_c_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)) ) )).

fof(fact_UnI2,axiom,(
    ! [V_A_2,V_B_2,V_c_2,T_a] :
      ( c_member(T_a,V_c_2,V_B_2)
     => c_member(T_a,V_c_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)) ) )).

fof(fact_insertI1,axiom,(
    ! [V_B_2,V_a_2,T_a] : c_member(T_a,V_a_2,c_Set_Oinsert(T_a,V_a_2,V_B_2)) )).

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

fof(fact_insert__ident,axiom,(
    ! [V_B_2,V_A_2,V_xa_2,T_a] :
      ( ~ c_member(T_a,V_xa_2,V_A_2)
     => ( ~ c_member(T_a,V_xa_2,V_B_2)
       => ( c_Set_Oinsert(T_a,V_xa_2,V_A_2) = c_Set_Oinsert(T_a,V_xa_2,V_B_2)
        <=> V_A_2 = V_B_2 ) ) ) )).

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

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

fof(fact_rtrancl__Un__rtrancl,axiom,(
    ! [V_S_2,V_R_2,T_a] : c_Transitive__Closure_Ortrancl(T_a,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),c_Transitive__Closure_Ortrancl(T_a,V_R_2),c_Transitive__Closure_Ortrancl(T_a,V_S_2))) = c_Transitive__Closure_Ortrancl(T_a,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_R_2,V_S_2)) )).

fof(fact_converse__Un,axiom,(
    ! [V_s_2,V_r_2,T_a,T_b] : c_Relation_Oconverse(T_b,T_a,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_a),tc_HOL_Obool),V_r_2,V_s_2)) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),c_Relation_Oconverse(T_b,T_a,V_r_2),c_Relation_Oconverse(T_b,T_a,V_s_2)) )).

fof(fact_in__rtrancl__UnI,axiom,(
    ! [V_S_2,V_R_2,V_xa_2,T_a] :
      ( ( c_member(tc_prod(T_a,T_a),V_xa_2,c_Transitive__Closure_Ortrancl(T_a,V_R_2))
        | c_member(tc_prod(T_a,T_a),V_xa_2,c_Transitive__Closure_Ortrancl(T_a,V_S_2)) )
     => c_member(tc_prod(T_a,T_a),V_xa_2,c_Transitive__Closure_Ortrancl(T_a,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_R_2,V_S_2))) ) )).

fof(fact_rtrancl__r__diff__Id,axiom,(
    ! [V_r_2,T_a] : c_Transitive__Closure_Ortrancl(T_a,c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,c_Relation_OId(T_a))) = c_Transitive__Closure_Ortrancl(T_a,V_r_2) )).

fof(fact_rtrancl__reflcl,axiom,(
    ! [V_R_2,T_a] : c_Transitive__Closure_Ortrancl(T_a,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_R_2,c_Relation_OId(T_a))) = c_Transitive__Closure_Ortrancl(T_a,V_R_2) )).

fof(fact_rtrancl__reflcl__absorb,axiom,(
    ! [V_R_2,T_a] : c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),c_Transitive__Closure_Ortrancl(T_a,V_R_2),c_Relation_OId(T_a)) = c_Transitive__Closure_Ortrancl(T_a,V_R_2) )).

fof(fact_antisym__reflcl,axiom,(
    ! [V_r_2,T_a] :
      ( c_Relation_Oantisym(T_a,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,c_Relation_OId(T_a)))
    <=> c_Relation_Oantisym(T_a,V_r_2) ) )).

fof(fact_total__on__diff__Id,axiom,(
    ! [V_r_2,V_A_2,T_a] :
      ( c_Relation_Ototal__on(T_a,V_A_2,c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,c_Relation_OId(T_a)))
    <=> c_Relation_Ototal__on(T_a,V_A_2,V_r_2) ) )).

fof(fact_minus__apply,axiom,(
    ! [V_xa_2,V_B_2,V_A_2,T_b,T_a] :
      ( class_Groups_Ominus(T_a)
     => hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,T_a),V_A_2,V_B_2),V_xa_2) = c_Groups_Ominus__class_Ominus(T_a,hAPP(V_A_2,V_xa_2),hAPP(V_B_2,V_xa_2)) ) )).

fof(fact_sup__apply,axiom,(
    ! [V_xa_2,V_g_2,V_f_2,T_b,T_a] :
      ( class_Lattices_Olattice(T_a)
     => hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,T_a),V_f_2,V_g_2),V_xa_2) = c_Lattices_Osemilattice__sup__class_Osup(T_a,hAPP(V_f_2,V_xa_2),hAPP(V_g_2,V_xa_2)) ) )).

fof(fact_rtrancl__Un__separator__converseE,axiom,(
    ! [V_Q_2,V_P_2,V_b_2,V_a_2,T_a] :
      ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Ortrancl(T_a,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_P_2,V_Q_2)))
     => ( ! [B_x] :
            ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_x,V_b_2),c_Transitive__Closure_Ortrancl(T_a,V_P_2))
           => ! [B_y] :
                ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_y,B_x),V_Q_2)
               => B_y = B_x ) )
       => c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Ortrancl(T_a,V_P_2)) ) ) )).

fof(fact_rtrancl__Un__separatorE,axiom,(
    ! [V_Q_2,V_P_2,V_b_2,V_a_2,T_a] :
      ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Ortrancl(T_a,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_P_2,V_Q_2)))
     => ( ! [B_x] :
            ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,B_x),c_Transitive__Closure_Ortrancl(T_a,V_P_2))
           => ! [B_y] :
                ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_x,B_y),V_Q_2)
               => B_x = B_y ) )
       => c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),c_Transitive__Closure_Ortrancl(T_a,V_P_2)) ) ) )).

fof(fact_trans__diff__Id,axiom,(
    ! [V_r_2,T_a] :
      ( c_Relation_Otrans(T_a,V_r_2)
     => ( c_Relation_Oantisym(T_a,V_r_2)
       => c_Relation_Otrans(T_a,c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,c_Relation_OId(T_a))) ) ) )).

fof(fact_wf__insert,axiom,(
    ! [V_r_2,V_xa_2,V_ya_2,T_a] :
      ( c_Wellfounded_Owf(T_a,c_Set_Oinsert(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_ya_2,V_xa_2),V_r_2))
    <=> ( c_Wellfounded_Owf(T_a,V_r_2)
        & ~ c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_ya_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)) ) ) )).

fof(fact_wf__inv__image,axiom,(
    ! [V_f_2,T_b,V_r_2,T_a] :
      ( c_Wellfounded_Owf(T_a,V_r_2)
     => c_Wellfounded_Owf(T_b,hAPP(c_Relation_Oinv__image(T_a,T_b,V_r_2),V_f_2)) ) )).

fof(fact_wf__lex__prod,axiom,(
    ! [V_rb_2,T_b,V_ra_2,T_a] :
      ( c_Wellfounded_Owf(T_a,V_ra_2)
     => ( c_Wellfounded_Owf(T_b,V_rb_2)
       => c_Wellfounded_Owf(tc_prod(T_a,T_b),c_Wellfounded_Olex__prod(T_a,T_b,V_ra_2,V_rb_2)) ) ) )).

fof(fact_trans__lex__prod,axiom,(
    ! [V_R2_2,T_b,V_R1_2,T_a] :
      ( c_Relation_Otrans(T_a,V_R1_2)
     => ( c_Relation_Otrans(T_b,V_R2_2)
       => c_Relation_Otrans(tc_prod(T_a,T_b),c_Wellfounded_Olex__prod(T_a,T_b,V_R1_2,V_R2_2)) ) ) )).

fof(fact_wf__trancl,axiom,(
    ! [V_r_2,T_a] :
      ( c_Wellfounded_Owf(T_a,V_r_2)
     => c_Wellfounded_Owf(T_a,c_Transitive__Closure_Otrancl(T_a,V_r_2)) ) )).

fof(fact_wf__acyclic,axiom,(
    ! [V_r_2,T_a] :
      ( c_Wellfounded_Owf(T_a,V_r_2)
     => c_Wellfounded_Oacyclic(T_a,V_r_2) ) )).

fof(fact_trans__rtrancl,axiom,(
    ! [V_r_2,T_a] : c_Relation_Otrans(T_a,c_Transitive__Closure_Ortrancl(T_a,V_r_2)) )).

fof(fact_trancl__id,axiom,(
    ! [V_r_2,T_a] :
      ( c_Relation_Otrans(T_a,V_r_2)
     => c_Transitive__Closure_Otrancl(T_a,V_r_2) = V_r_2 ) )).

fof(fact_trans__trancl,axiom,(
    ! [V_r_2,T_a] : c_Relation_Otrans(T_a,c_Transitive__Closure_Otrancl(T_a,V_r_2)) )).

fof(fact_trans__converse,axiom,(
    ! [V_r_2,T_a] :
      ( c_Relation_Otrans(T_a,c_Relation_Oconverse(T_a,T_a,V_r_2))
    <=> c_Relation_Otrans(T_a,V_r_2) ) )).

fof(fact_trans__Id,axiom,(
    ! [T_a] : c_Relation_Otrans(T_a,c_Relation_OId(T_a)) )).

fof(fact_trans__Id__on,axiom,(
    ! [V_A_2,T_a] : c_Relation_Otrans(T_a,c_Relation_OId__on(T_a,V_A_2)) )).

fof(fact_trans__inv__image,axiom,(
    ! [V_f_2,T_b,V_r_2,T_a] :
      ( c_Relation_Otrans(T_a,V_r_2)
     => c_Relation_Otrans(T_b,hAPP(c_Relation_Oinv__image(T_a,T_b,V_r_2),V_f_2)) ) )).

fof(fact_wf__irrefl,axiom,(
    ! [V_a_2,V_r_2,T_a] :
      ( c_Wellfounded_Owf(T_a,V_r_2)
     => ~ c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_a_2),V_r_2) ) )).

fof(fact_wf__asym,axiom,(
    ! [V_xa_2,V_a_2,V_r_2,T_a] :
      ( c_Wellfounded_Owf(T_a,V_r_2)
     => ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_xa_2),V_r_2)
       => ~ c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_a_2),V_r_2) ) ) )).

fof(fact_wf__not__sym,axiom,(
    ! [V_xa_2,V_a_2,V_r_2,T_a] :
      ( c_Wellfounded_Owf(T_a,V_r_2)
     => ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_xa_2),V_r_2)
       => ~ c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_a_2),V_r_2) ) ) )).

fof(fact_wf__not__refl,axiom,(
    ! [V_a_2,V_r_2,T_a] :
      ( c_Wellfounded_Owf(T_a,V_r_2)
     => ~ c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_a_2),V_r_2) ) )).

fof(fact_acc__wfD,axiom,(
    ! [V_xa_2,V_r_2,T_a] :
      ( c_Wellfounded_Owf(T_a,V_r_2)
     => c_member(T_a,V_xa_2,c_Wellfounded_Oacc(T_a,V_r_2)) ) )).

fof(fact_wf__acc__iff,axiom,(
    ! [V_r_2,T_a] :
      ( c_Wellfounded_Owf(T_a,V_r_2)
    <=> ! [B_x] : c_member(T_a,B_x,c_Wellfounded_Oacc(T_a,V_r_2)) ) )).

fof(fact_wf__converse__trancl,axiom,(
    ! [V_r_2,T_a] :
      ( c_Wellfounded_Owf(T_a,c_Relation_Oconverse(T_a,T_a,V_r_2))
     => c_Wellfounded_Owf(T_a,c_Relation_Oconverse(T_a,T_a,c_Transitive__Closure_Otrancl(T_a,V_r_2))) ) )).

fof(fact_transD,axiom,(
    ! [V_c_2,V_b_2,V_a_2,V_r_2,T_a] :
      ( c_Relation_Otrans(T_a,V_r_2)
     => ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),V_r_2)
       => ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_b_2,V_c_2),V_r_2)
         => c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_c_2),V_r_2) ) ) ) )).

fof(fact_trans__def,axiom,(
    ! [V_r_2,T_a] :
      ( c_Relation_Otrans(T_a,V_r_2)
    <=> ! [B_x,B_y] :
          ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_x,B_y),V_r_2)
         => ! [B_z] :
              ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_y,B_z),V_r_2)
             => c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_x,B_z),V_r_2) ) ) ) )).

fof(fact_trans__reflclI,axiom,(
    ! [V_r_2,T_a] :
      ( c_Relation_Otrans(T_a,V_r_2)
     => c_Relation_Otrans(T_a,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,c_Relation_OId(T_a))) ) )).

fof(fact_sup_Oidem,axiom,(
    ! [V_a,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => 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)
     => 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)
     => c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,V_b) = 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)
     => c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_y) = 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)
     => c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_y) = 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)
     => c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,V_b)) = 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)
     => c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_y)) = 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)
     => c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_y)) = 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)
     => c_Lattices_Osemilattice__sup__class_Osup(T_a,V_b,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,V_c)) = c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,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)
     => c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_y,V_z)) = c_Lattices_Osemilattice__sup__class_Osup(T_a,V_y,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)
     => c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_y,V_z)) = c_Lattices_Osemilattice__sup__class_Osup(T_a,V_y,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)
     => c_Lattices_Osemilattice__sup__class_Osup(T_a,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,V_b),V_c) = c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,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)
     => c_Lattices_Osemilattice__sup__class_Osup(T_a,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_y),V_z) = c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,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)
     => c_Lattices_Osemilattice__sup__class_Osup(T_a,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_y),V_z) = c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_y,V_z)) ) )).

fof(fact_strict__linear__order__on__def,axiom,(
    ! [V_r_2,V_A_2,T_a] :
      ( c_Order__Relation_Ostrict__linear__order__on(T_a,V_A_2,V_r_2)
    <=> ( c_Relation_Otrans(T_a,V_r_2)
        & c_Relation_Oirrefl(T_a,V_r_2)
        & c_Relation_Ototal__on(T_a,V_A_2,V_r_2) ) ) )).

fof(fact_wf__measure,axiom,(
    ! [V_f_2,T_a] : c_Wellfounded_Owf(T_a,hAPP(c_Wellfounded_Omeasure(T_a),V_f_2)) )).

fof(fact_tfl__wfrec,axiom,(
    ! [T_b,T_a,B_M,B_R] :
      ( c_Wellfounded_Owf(T_a,B_R)
     => ! [B_x] : hAPP(c_Recdef_Owfrec(T_a,T_b,B_R,B_M),B_x) = hAPP(hAPP(B_M,c_Recdef_Ocut(T_a,T_b,c_Recdef_Owfrec(T_a,T_b,B_R,B_M),B_R,B_x)),B_x) ) )).

fof(fact_wfrec,axiom,(
    ! [V_a_2,V_H_2,T_b,V_r_2,T_a] :
      ( c_Wellfounded_Owf(T_a,V_r_2)
     => hAPP(c_Recdef_Owfrec(T_a,T_b,V_r_2,V_H_2),V_a_2) = hAPP(hAPP(V_H_2,c_Recdef_Ocut(T_a,T_b,c_Recdef_Owfrec(T_a,T_b,V_r_2,V_H_2),V_r_2,V_a_2)),V_a_2) ) )).

fof(fact_def__wfrec,axiom,(
    ! [V_a_2,V_H_2,V_r_2,T_b,T_a,V_f_2] :
      ( V_f_2 = c_Recdef_Owfrec(T_a,T_b,V_r_2,V_H_2)
     => ( c_Wellfounded_Owf(T_a,V_r_2)
       => hAPP(V_f_2,V_a_2) = hAPP(hAPP(V_H_2,c_Recdef_Ocut(T_a,T_b,V_f_2,V_r_2,V_a_2)),V_a_2) ) ) )).

fof(fact_strict__linear__order__on__diff__Id,axiom,(
    ! [V_r_2,V_A_2,T_a] :
      ( c_Order__Relation_Olinear__order__on(T_a,V_A_2,V_r_2)
     => c_Order__Relation_Ostrict__linear__order__on(T_a,V_A_2,c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,c_Relation_OId(T_a))) ) )).

fof(fact_wf__same__fst,axiom,(
    ! [T_a,V_R_2,T_b,V_P_2] :
      ( ! [B_x] :
          ( hBOOL(hAPP(V_P_2,B_x))
         => c_Wellfounded_Owf(T_b,hAPP(V_R_2,B_x)) )
     => c_Wellfounded_Owf(tc_prod(T_a,T_b),c_Recdef_Osame__fst(T_a,T_b,V_P_2,V_R_2)) ) )).

fof(fact_preorder__on__def,axiom,(
    ! [V_r_2,V_A_2,T_a] :
      ( c_Order__Relation_Opreorder__on(T_a,V_A_2,V_r_2)
    <=> ( c_Relation_Orefl__on(T_a,V_A_2,V_r_2)
        & c_Relation_Otrans(T_a,V_r_2) ) ) )).

fof(fact_wf__eq__minimal,axiom,(
    ! [V_r_2,T_a] :
      ( c_Wellfounded_Owf(T_a,V_r_2)
    <=> ! [B_Q] :
          ( ? [B_x] : c_member(T_a,B_x,B_Q)
         => ? [B_x] :
              ( c_member(T_a,B_x,B_Q)
              & ! [B_y] :
                  ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_y,B_x),V_r_2)
                 => ~ c_member(T_a,B_y,B_Q) ) ) ) ) )).

fof(fact_preorder__on__converse,axiom,(
    ! [V_r_2,V_A_2,T_a] :
      ( c_Order__Relation_Opreorder__on(T_a,V_A_2,c_Relation_Oconverse(T_a,T_a,V_r_2))
    <=> c_Order__Relation_Opreorder__on(T_a,V_A_2,V_r_2) ) )).

fof(fact_linear__order__on__converse,axiom,(
    ! [V_r_2,V_A_2,T_a] :
      ( c_Order__Relation_Olinear__order__on(T_a,V_A_2,c_Relation_Oconverse(T_a,T_a,V_r_2))
    <=> c_Order__Relation_Olinear__order__on(T_a,V_A_2,V_r_2) ) )).

fof(fact_well__order__on__def,axiom,(
    ! [V_r_2,V_A_2,T_a] :
      ( c_Order__Relation_Owell__order__on(T_a,V_A_2,V_r_2)
    <=> ( c_Order__Relation_Olinear__order__on(T_a,V_A_2,V_r_2)
        & c_Wellfounded_Owf(T_a,c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,c_Relation_OId(T_a))) ) ) )).

fof(fact_linear__order__on__def,axiom,(
    ! [V_r_2,V_A_2,T_a] :
      ( c_Order__Relation_Olinear__order__on(T_a,V_A_2,V_r_2)
    <=> ( c_Order__Relation_Opartial__order__on(T_a,V_A_2,V_r_2)
        & c_Relation_Ototal__on(T_a,V_A_2,V_r_2) ) ) )).

fof(fact_partial__order__on__def,axiom,(
    ! [V_r_2,V_A_2,T_a] :
      ( c_Order__Relation_Opartial__order__on(T_a,V_A_2,V_r_2)
    <=> ( c_Order__Relation_Opreorder__on(T_a,V_A_2,V_r_2)
        & c_Relation_Oantisym(T_a,V_r_2) ) ) )).

fof(fact_wf__in__rel,axiom,(
    ! [V_R_2,T_a] :
      ( c_Wellfounded_Owf(T_a,V_R_2)
     => c_Wellfounded_OwfP(T_a,c_FunDef_Oin__rel(T_a,T_a,V_R_2)) ) )).

fof(fact_partial__order__on__converse,axiom,(
    ! [V_r_2,V_A_2,T_a] :
      ( c_Order__Relation_Opartial__order__on(T_a,V_A_2,c_Relation_Oconverse(T_a,T_a,V_r_2))
    <=> c_Order__Relation_Opartial__order__on(T_a,V_A_2,V_r_2) ) )).

fof(fact_acc__wfI,axiom,(
    ! [V_r_2,T_a] :
      ( ! [B_x] : c_member(T_a,B_x,c_Wellfounded_Oacc(T_a,V_r_2))
     => c_Wellfounded_Owf(T_a,V_r_2) ) )).

fof(fact_Field__def,axiom,(
    ! [V_r_2,T_a] : c_Relation_OField(T_a,V_r_2) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),c_Relation_ODomain(T_a,T_a,V_r_2),c_Relation_ORange(T_a,T_a,V_r_2)) )).

fof(fact_in__measure,axiom,(
    ! [V_f_2,V_ya_2,V_xa_2,T_a] :
      ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_ya_2),hAPP(c_Wellfounded_Omeasure(T_a),V_f_2))
    <=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,hAPP(V_f_2,V_xa_2),hAPP(V_f_2,V_ya_2)) ) )).

fof(fact_transI,axiom,(
    ! [V_r_2,T_a] :
      ( ! [B_x,B_y] :
          ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_x,B_y),V_r_2)
         => ! [B_z] :
              ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_y,B_z),V_r_2)
             => c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_x,B_z),V_r_2) ) )
     => c_Relation_Otrans(T_a,V_r_2) ) )).

fof(fact_Field__converse,axiom,(
    ! [V_r_2,T_a] : c_Relation_OField(T_a,c_Relation_Oconverse(T_a,T_a,V_r_2)) = c_Relation_OField(T_a,V_r_2) )).

fof(fact_less__supI1,axiom,(
    ! [V_b,V_a,V_x,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => ( c_Orderings_Oord__class_Oless(T_a,V_x,V_a)
       => c_Orderings_Oord__class_Oless(T_a,V_x,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,V_b)) ) ) )).

fof(fact_less__supI2,axiom,(
    ! [V_a,V_b,V_x,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => ( c_Orderings_Oord__class_Oless(T_a,V_x,V_b)
       => c_Orderings_Oord__class_Oless(T_a,V_x,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,V_b)) ) ) )).

fof(fact_Field__Un,axiom,(
    ! [V_s_2,V_r_2,T_a] : c_Relation_OField(T_a,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,V_s_2)) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),c_Relation_OField(T_a,V_r_2),c_Relation_OField(T_a,V_s_2)) )).

fof(fact_mlex__less,axiom,(
    ! [V_R_2,T_a,V_ya_2,V_xa_2,V_f_2] :
      ( c_Orderings_Oord__class_Oless(tc_Nat_Onat,hAPP(V_f_2,V_xa_2),hAPP(V_f_2,V_ya_2))
     => c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_ya_2),c_Wellfounded_Omlex__prod(T_a,V_f_2,V_R_2)) ) )).

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

fof(fact_Field__insert,axiom,(
    ! [V_r_2,V_b_2,V_a_2,T_a] : c_Relation_OField(T_a,c_Set_Oinsert(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),V_r_2)) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),c_Set_Oinsert(T_a,V_a_2,c_Set_Oinsert(T_a,V_b_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))),c_Relation_OField(T_a,V_r_2)) )).

fof(fact_measure__def,axiom,(
    ! [T_a] : c_Wellfounded_Omeasure(T_a) = c_Relation_Oinv__image(tc_Nat_Onat,T_a,c_Wellfounded_Oless__than) )).

fof(fact_wf__empty,axiom,(
    ! [T_a] : c_Wellfounded_Owf(T_a,c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool))) )).

fof(fact_trans__less__than,axiom,(
    c_Relation_Otrans(tc_Nat_Onat,c_Wellfounded_Oless__than) )).

fof(fact_wf__less__than,axiom,(
    c_Wellfounded_Owf(tc_Nat_Onat,c_Wellfounded_Oless__than) )).

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

fof(fact_Field__empty,axiom,(
    ! [T_a] : c_Relation_OField(T_a,c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) )).

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

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

fof(fact_lnear__order__on__empty,axiom,(
    ! [T_a] : c_Order__Relation_Olinear__order__on(T_a,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool))) )).

fof(fact_preorder__on__empty,axiom,(
    ! [T_a] : c_Order__Relation_Opreorder__on(T_a,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool))) )).

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

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

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

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

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

fof(fact_doubleton__eq__iff,axiom,(
    ! [V_d_2,V_c_2,V_b_2,V_a_2,T_a] :
      ( c_Set_Oinsert(T_a,V_a_2,c_Set_Oinsert(T_a,V_b_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) = c_Set_Oinsert(T_a,V_c_2,c_Set_Oinsert(T_a,V_d_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,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_insert__not__empty,axiom,(
    ! [V_A_2,V_a_2,T_a] : c_Set_Oinsert(T_a,V_a_2,V_A_2) != c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) )).

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

fof(fact_refl__on__empty,axiom,(
    ! [T_a] : c_Relation_Orefl__on(T_a,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool))) )).

fof(fact_Range__empty,axiom,(
    ! [T_a,T_b] : c_Relation_ORange(T_b,T_a,c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_b,T_a),tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) )).

fof(fact_Range__empty__iff,axiom,(
    ! [V_r_2,T_a,T_b] :
      ( c_Relation_ORange(T_b,T_a,V_r_2) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))
    <=> V_r_2 = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_b,T_a),tc_HOL_Obool)) ) )).

fof(fact_Sigma__empty1,axiom,(
    ! [V_B_2,T_b,T_a] : c_Product__Type_OSigma(T_a,T_b,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)),V_B_2) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool)) )).

fof(fact_Id__on__empty,axiom,(
    ! [T_a] : c_Relation_OId__on(T_a,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool)) )).

fof(fact_Domain__empty__iff,axiom,(
    ! [V_r_2,T_b,T_a] :
      ( c_Relation_ODomain(T_a,T_b,V_r_2) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))
    <=> V_r_2 = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool)) ) )).

fof(fact_Domain__empty,axiom,(
    ! [T_b,T_a] : c_Relation_ODomain(T_a,T_b,c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) )).

fof(fact_trancl__empty,axiom,(
    ! [T_a] : c_Transitive__Closure_Otrancl(T_a,c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool)) )).

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

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

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

fof(fact_Un__empty,axiom,(
    ! [V_B_2,V_A_2,T_a] :
      ( c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))
    <=> ( V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))
        & V_B_2 = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) ) ) )).

fof(fact_Un__empty__right,axiom,(
    ! [V_A_2,T_a] : c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) = V_A_2 )).

fof(fact_Un__empty__left,axiom,(
    ! [V_B_2,T_a] : c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)),V_B_2) = V_B_2 )).

fof(fact_antisym__empty,axiom,(
    ! [T_a] : c_Relation_Oantisym(T_a,c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool))) )).

fof(fact_partial__order__on__empty,axiom,(
    ! [T_a] : c_Order__Relation_Opartial__order__on(T_a,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool))) )).

fof(fact_well__order__on__empty,axiom,(
    ! [T_a] : c_Order__Relation_Owell__order__on(T_a,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool))) )).

fof(fact_total__on__empty,axiom,(
    ! [V_r_2,T_a] : c_Relation_Ototal__on(T_a,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)),V_r_2) )).

fof(fact_sup__bot__left,axiom,(
    ! [V_x,T_a] :
      ( class_Lattices_Obounded__lattice__bot(T_a)
     => c_Lattices_Osemilattice__sup__class_Osup(T_a,c_Orderings_Obot__class_Obot(T_a),V_x) = V_x ) )).

fof(fact_sup__bot__right,axiom,(
    ! [V_x,T_a] :
      ( class_Lattices_Obounded__lattice__bot(T_a)
     => c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,c_Orderings_Obot__class_Obot(T_a)) = V_x ) )).

fof(fact_sup__eq__bot__iff,axiom,(
    ! [V_ya_2,V_xa_2,T_a] :
      ( class_Lattices_Obounded__lattice__bot(T_a)
     => ( c_Lattices_Osemilattice__sup__class_Osup(T_a,V_xa_2,V_ya_2) = c_Orderings_Obot__class_Obot(T_a)
      <=> ( V_xa_2 = c_Orderings_Obot__class_Obot(T_a)
          & V_ya_2 = c_Orderings_Obot__class_Obot(T_a) ) ) ) )).

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

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

fof(fact_insert__is__Un,axiom,(
    ! [V_A_2,V_a_2,T_a] : c_Set_Oinsert(T_a,V_a_2,V_A_2) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),c_Set_Oinsert(T_a,V_a_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))),V_A_2) )).

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

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

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

fof(fact_less__than__iff,axiom,(
    ! [V_ya_2,V_xa_2] :
      ( c_member(tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat,V_xa_2,V_ya_2),c_Wellfounded_Oless__than)
    <=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_xa_2,V_ya_2) ) )).

fof(fact_rtrancl__empty,axiom,(
    ! [T_a] : c_Transitive__Closure_Ortrancl(T_a,c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool))) = c_Relation_OId(T_a) )).

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

fof(fact_Diff__insert__absorb,axiom,(
    ! [V_A_2,V_xa_2,T_a] :
      ( ~ c_member(T_a,V_xa_2,V_A_2)
     => c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),c_Set_Oinsert(T_a,V_xa_2,V_A_2),c_Set_Oinsert(T_a,V_xa_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) = V_A_2 ) )).

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

fof(fact_wf__mlex,axiom,(
    ! [V_f_2,V_R_2,T_a] :
      ( c_Wellfounded_Owf(T_a,V_R_2)
     => c_Wellfounded_Owf(T_a,c_Wellfounded_Omlex__prod(T_a,V_f_2,V_R_2)) ) )).

fof(fact_the__elem__eq,axiom,(
    ! [V_xa_2,T_a] : c_Set_Othe__elem(T_a,c_Set_Oinsert(T_a,V_xa_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) = V_xa_2 )).

fof(fact_fold__graph_H_Ointros_I2_J,axiom,(
    ! [V_ya_2,V_z_2,V_f_2,T_b,V_A_2,V_xa_2,T_a] :
      ( c_member(T_a,V_xa_2,V_A_2)
     => ( c_Nitpick_Ofold__graph_H(T_a,T_b,V_f_2,V_z_2,c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Set_Oinsert(T_a,V_xa_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))),V_ya_2)
       => c_Nitpick_Ofold__graph_H(T_a,T_b,V_f_2,V_z_2,V_A_2,hAPP(hAPP(V_f_2,V_xa_2),V_ya_2)) ) ) )).

fof(fact_bot__apply,axiom,(
    ! [V_xa_2,T_b,T_a] :
      ( class_Orderings_Obot(T_a)
     => hAPP(c_Orderings_Obot__class_Obot(tc_fun(T_b,T_a)),V_xa_2) = c_Orderings_Obot__class_Obot(T_a) ) )).

fof(fact_Refl__antisym__eq__Image1__Image1__iff,axiom,(
    ! [V_b_2,V_a_2,V_r_2,T_a] :
      ( c_Relation_Orefl__on(T_a,c_Relation_OField(T_a,V_r_2),V_r_2)
     => ( c_Relation_Oantisym(T_a,V_r_2)
       => ( c_member(T_a,V_a_2,c_Relation_OField(T_a,V_r_2))
         => ( c_member(T_a,V_b_2,c_Relation_OField(T_a,V_r_2))
           => ( c_Relation_OImage(T_a,T_a,V_r_2,c_Set_Oinsert(T_a,V_a_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) = c_Relation_OImage(T_a,T_a,V_r_2,c_Set_Oinsert(T_a,V_b_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))))
            <=> V_a_2 = V_b_2 ) ) ) ) ) )).

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

fof(fact_Image__empty,axiom,(
    ! [V_R_2,T_a,T_b] : c_Relation_OImage(T_b,T_a,V_R_2,c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) )).

fof(fact_Image__Id,axiom,(
    ! [V_A_2,T_a] : c_Relation_OImage(T_a,T_a,c_Relation_OId(T_a),V_A_2) = V_A_2 )).

fof(fact_Image__Un,axiom,(
    ! [V_B_2,V_A_2,V_R_2,T_a,T_b] : c_Relation_OImage(T_b,T_a,V_R_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool),V_A_2,V_B_2)) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),c_Relation_OImage(T_b,T_a,V_R_2,V_A_2),c_Relation_OImage(T_b,T_a,V_R_2,V_B_2)) )).

fof(fact_Un__Image,axiom,(
    ! [V_A_2,V_S_2,V_R_2,T_a,T_b] : c_Relation_OImage(T_b,T_a,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_a),tc_HOL_Obool),V_R_2,V_S_2),V_A_2) = c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),c_Relation_OImage(T_b,T_a,V_R_2,V_A_2),c_Relation_OImage(T_b,T_a,V_S_2,V_A_2)) )).

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

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

fof(fact_rev__ImageI,axiom,(
    ! [V_r_2,V_b_2,T_b,V_A_2,V_a_2,T_a] :
      ( c_member(T_a,V_a_2,V_A_2)
     => ( c_member(tc_prod(T_a,T_b),c_Product__Type_OPair(T_a,T_b,V_a_2,V_b_2),V_r_2)
       => c_member(T_b,V_b_2,c_Relation_OImage(T_a,T_b,V_r_2,V_A_2)) ) ) )).

fof(fact_Image__iff,axiom,(
    ! [V_A_2,V_r_2,T_b,V_b_2,T_a] :
      ( c_member(T_a,V_b_2,c_Relation_OImage(T_b,T_a,V_r_2,V_A_2))
    <=> ? [B_x] :
          ( c_member(T_b,B_x,V_A_2)
          & c_member(tc_prod(T_b,T_a),c_Product__Type_OPair(T_b,T_a,B_x,V_b_2),V_r_2) ) ) )).

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

fof(fact_linorder__neq__iff,axiom,(
    ! [V_ya_2,V_xa_2,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => ( V_xa_2 != V_ya_2
      <=> ( c_Orderings_Oord__class_Oless(T_a,V_xa_2,V_ya_2)
          | c_Orderings_Oord__class_Oless(T_a,V_ya_2,V_xa_2) ) ) ) )).

fof(fact_not__less__iff__gr__or__eq,axiom,(
    ! [V_ya_2,V_xa_2,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => ( ~ c_Orderings_Oord__class_Oless(T_a,V_xa_2,V_ya_2)
      <=> ( c_Orderings_Oord__class_Oless(T_a,V_ya_2,V_xa_2)
          | V_xa_2 = V_ya_2 ) ) ) )).

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

fof(fact_linorder__antisym__conv3,axiom,(
    ! [V_xa_2,V_ya_2,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => ( ~ c_Orderings_Oord__class_Oless(T_a,V_ya_2,V_xa_2)
       => ( ~ c_Orderings_Oord__class_Oless(T_a,V_xa_2,V_ya_2)
        <=> V_xa_2 = V_ya_2 ) ) ) )).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

fof(fact_Image__singleton__iff,axiom,(
    ! [V_a_2,V_r_2,T_b,V_b_2,T_a] :
      ( c_member(T_a,V_b_2,c_Relation_OImage(T_b,T_a,V_r_2,c_Set_Oinsert(T_b,V_a_2,c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))
    <=> c_member(tc_prod(T_b,T_a),c_Product__Type_OPair(T_b,T_a,V_a_2,V_b_2),V_r_2) ) )).

fof(fact_Partial__order__eq__Image1__Image1__iff,axiom,(
    ! [V_b_2,V_a_2,V_r_2,T_a] :
      ( c_Order__Relation_Opartial__order__on(T_a,c_Relation_OField(T_a,V_r_2),V_r_2)
     => ( c_member(T_a,V_a_2,c_Relation_OField(T_a,V_r_2))
       => ( c_member(T_a,V_b_2,c_Relation_OField(T_a,V_r_2))
         => ( c_Relation_OImage(T_a,T_a,V_r_2,c_Set_Oinsert(T_a,V_a_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))) = c_Relation_OImage(T_a,T_a,V_r_2,c_Set_Oinsert(T_a,V_b_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))))
          <=> V_a_2 = V_b_2 ) ) ) ) )).

fof(fact_less__eq,axiom,(
    ! [V_n_2,V_m_2] :
      ( c_member(tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat,V_m_2,V_n_2),c_Transitive__Closure_Otrancl(tc_Nat_Onat,c_Wellfounded_Opred__nat))
    <=> c_Orderings_Oord__class_Oless(tc_Nat_Onat,V_m_2,V_n_2) ) )).

fof(fact_subset__Image1__Image1__iff,axiom,(
    ! [V_b_2,V_a_2,V_r_2,T_a] :
      ( c_Order__Relation_Opreorder__on(T_a,c_Relation_OField(T_a,V_r_2),V_r_2)
     => ( c_member(T_a,V_a_2,c_Relation_OField(T_a,V_r_2))
       => ( c_member(T_a,V_b_2,c_Relation_OField(T_a,V_r_2))
         => ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Relation_OImage(T_a,T_a,V_r_2,c_Set_Oinsert(T_a,V_a_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))),c_Relation_OImage(T_a,T_a,V_r_2,c_Set_Oinsert(T_a,V_b_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))))
          <=> c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_b_2,V_a_2),V_r_2) ) ) ) ) )).

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_b,T_a] :
      ( c_Nitpick_Ofold__graph_H(T_a,T_b,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_a,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)
            & c_member(T_a,B_x,V_a2_2)
            & c_Nitpick_Ofold__graph_H(T_a,T_b,V_a3_2,V_a4_2,c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_a2_2,c_Set_Oinsert(T_a,B_x,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))),B_y) ) ) ) )).

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

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

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

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

fof(fact_linorder__not__less,axiom,(
    ! [V_ya_2,V_xa_2,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => ( ~ c_Orderings_Oord__class_Oless(T_a,V_xa_2,V_ya_2)
      <=> c_Orderings_Oord__class_Oless__eq(T_a,V_ya_2,V_xa_2) ) ) )).

fof(fact_linorder__not__le,axiom,(
    ! [V_ya_2,V_xa_2,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => ( ~ c_Orderings_Oord__class_Oless__eq(T_a,V_xa_2,V_ya_2)
      <=> c_Orderings_Oord__class_Oless(T_a,V_ya_2,V_xa_2) ) ) )).

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

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

fof(fact_order__less__le,axiom,(
    ! [V_ya_2,V_xa_2,T_a] :
      ( class_Orderings_Oorder(T_a)
     => ( c_Orderings_Oord__class_Oless(T_a,V_xa_2,V_ya_2)
      <=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_xa_2,V_ya_2)
          & V_xa_2 != V_ya_2 ) ) ) )).

fof(fact_less__le__not__le,axiom,(
    ! [V_ya_2,V_xa_2,T_a] :
      ( class_Orderings_Opreorder(T_a)
     => ( c_Orderings_Oord__class_Oless(T_a,V_xa_2,V_ya_2)
      <=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_xa_2,V_ya_2)
          & ~ c_Orderings_Oord__class_Oless__eq(T_a,V_ya_2,V_xa_2) ) ) ) )).

fof(fact_order__le__less,axiom,(
    ! [V_ya_2,V_xa_2,T_a] :
      ( class_Orderings_Oorder(T_a)
     => ( c_Orderings_Oord__class_Oless__eq(T_a,V_xa_2,V_ya_2)
      <=> ( c_Orderings_Oord__class_Oless(T_a,V_xa_2,V_ya_2)
          | V_xa_2 = V_ya_2 ) ) ) )).

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

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

fof(fact_linorder__antisym__conv1,axiom,(
    ! [V_ya_2,V_xa_2,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => ( ~ c_Orderings_Oord__class_Oless(T_a,V_xa_2,V_ya_2)
       => ( c_Orderings_Oord__class_Oless__eq(T_a,V_xa_2,V_ya_2)
        <=> V_xa_2 = V_ya_2 ) ) ) )).

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

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

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

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

fof(fact_linorder__antisym__conv2,axiom,(
    ! [V_ya_2,V_xa_2,T_a] :
      ( class_Orderings_Olinorder(T_a)
     => ( c_Orderings_Oord__class_Oless__eq(T_a,V_xa_2,V_ya_2)
       => ( ~ c_Orderings_Oord__class_Oless(T_a,V_xa_2,V_ya_2)
        <=> V_xa_2 = V_ya_2 ) ) ) )).

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

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

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

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

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

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

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

fof(fact_subset__iff__psubset__eq,axiom,(
    ! [V_B_2,V_A_2,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)
    <=> ( c_Orderings_Oord__class_Oless(tc_fun(T_a,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_a] :
      ( c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)
    <=> ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)
        & V_A_2 != V_B_2 ) ) )).

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

fof(fact_less__by__empty,axiom,(
    ! [V_B_2,T_a,V_A_2] :
      ( V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool))
     => c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_A_2,V_B_2) ) )).

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

fof(fact_wf__subset,axiom,(
    ! [V_p_2,V_r_2,T_a] :
      ( c_Wellfounded_Owf(T_a,V_r_2)
     => ( c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_p_2,V_r_2)
       => c_Wellfounded_Owf(T_a,V_p_2) ) ) )).

fof(fact_Image__mono,axiom,(
    ! [V_A_2,V_A_H_2,V_r_2,V_r_H_2,T_b,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),V_r_H_2,V_r_2)
     => ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_H_2,V_A_2)
       => c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool),c_Relation_OImage(T_a,T_b,V_r_H_2,V_A_H_2),c_Relation_OImage(T_a,T_b,V_r_2,V_A_2)) ) ) )).

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

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

fof(fact_order__eq__iff,axiom,(
    ! [V_ya_2,V_xa_2,T_a] :
      ( class_Orderings_Oorder(T_a)
     => ( V_xa_2 = V_ya_2
      <=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_xa_2,V_ya_2)
          & c_Orderings_Oord__class_Oless__eq(T_a,V_ya_2,V_xa_2) ) ) ) )).

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

fof(fact_le__funD,axiom,(
    ! [V_xa_2,V_g_2,V_f_2,T_a,T_b] :
      ( class_Orderings_Oord(T_b)
     => ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,T_b),V_f_2,V_g_2)
       => c_Orderings_Oord__class_Oless__eq(T_b,hAPP(V_f_2,V_xa_2),hAPP(V_g_2,V_xa_2)) ) ) )).

fof(fact_order__antisym__conv,axiom,(
    ! [V_xa_2,V_ya_2,T_a] :
      ( class_Orderings_Oorder(T_a)
     => ( c_Orderings_Oord__class_Oless__eq(T_a,V_ya_2,V_xa_2)
       => ( c_Orderings_Oord__class_Oless__eq(T_a,V_xa_2,V_ya_2)
        <=> V_xa_2 = V_ya_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
       => ( c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_c)
         => 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
       => ( c_Orderings_Oord__class_Oless__eq(T_a,V_c,V_b)
         => c_Orderings_Oord__class_Oless__eq(T_a,V_c,V_a) ) ) ) )).

fof(fact_ord__le__eq__trans,axiom,(
    ! [V_c,V_b,V_a,T_a] :
      ( class_Orderings_Oord(T_a)
     => ( c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_b)
       => ( V_b = V_c
         => 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)
     => ( c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_a)
       => ( V_b = V_c
         => 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)
     => ( c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y)
       => ( 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)
     => ( c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y)
       => ( c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_z)
         => 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)
     => ( c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x)
       => ( 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)
     => ( c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x)
       => ( c_Orderings_Oord__class_Oless__eq(T_a,V_z,V_y)
         => c_Orderings_Oord__class_Oless__eq(T_a,V_z,V_x) ) ) ) )).

fof(fact_le__funE,axiom,(
    ! [V_xa_2,V_g_2,V_f_2,T_a,T_b] :
      ( class_Orderings_Oord(T_b)
     => ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,T_b),V_f_2,V_g_2)
       => c_Orderings_Oord__class_Oless__eq(T_b,hAPP(V_f_2,V_xa_2),hAPP(V_g_2,V_xa_2)) ) ) )).

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

fof(fact_in__mono,axiom,(
    ! [V_xa_2,V_B_2,V_A_2,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)
     => ( c_member(T_a,V_xa_2,V_A_2)
       => c_member(T_a,V_xa_2,V_B_2) ) ) )).

fof(fact_set__rev__mp,axiom,(
    ! [V_B_2,V_A_2,V_xa_2,T_a] :
      ( c_member(T_a,V_xa_2,V_A_2)
     => ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)
       => c_member(T_a,V_xa_2,V_B_2) ) ) )).

fof(fact_set__mp,axiom,(
    ! [V_xa_2,V_B_2,V_A_2,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)
     => ( c_member(T_a,V_xa_2,V_A_2)
       => c_member(T_a,V_xa_2,V_B_2) ) ) )).

fof(fact_mono__Field,axiom,(
    ! [V_s_2,V_r_2,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,V_s_2)
     => c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Relation_OField(T_a,V_r_2),c_Relation_OField(T_a,V_s_2)) ) )).

fof(fact_single__valued__subset,axiom,(
    ! [V_s_2,V_r_2,T_b,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),V_r_2,V_s_2)
     => ( c_Relation_Osingle__valued(T_a,T_b,V_s_2)
       => c_Relation_Osingle__valued(T_a,T_b,V_r_2) ) ) )).

fof(fact_le__supE,axiom,(
    ! [V_x,V_b,V_a,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => ( c_Orderings_Oord__class_Oless__eq(T_a,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,V_b),V_x)
       => ~ ( c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_x)
           => ~ 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)
     => ( c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_c)
       => ( c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_d)
         => c_Orderings_Oord__class_Oless__eq(T_a,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,V_b),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)
     => ( c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x)
       => ( c_Orderings_Oord__class_Oless__eq(T_a,V_z,V_x)
         => c_Orderings_Oord__class_Oless__eq(T_a,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)
     => ( c_Orderings_Oord__class_Oless__eq(T_a,V_a,V_x)
       => ( c_Orderings_Oord__class_Oless__eq(T_a,V_b,V_x)
         => c_Orderings_Oord__class_Oless__eq(T_a,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)
     => ( c_Orderings_Oord__class_Oless__eq(T_a,V_y,V_x)
       => 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)
     => ( c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_y)
       => 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)
     => ( c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_b)
       => c_Orderings_Oord__class_Oless__eq(T_a,V_x,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)
     => ( c_Orderings_Oord__class_Oless__eq(T_a,V_x,V_a)
       => c_Orderings_Oord__class_Oless__eq(T_a,V_x,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_a,V_b)) ) ) )).

fof(fact_le__sup__iff,axiom,(
    ! [V_z_2,V_ya_2,V_xa_2,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => ( c_Orderings_Oord__class_Oless__eq(T_a,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_xa_2,V_ya_2),V_z_2)
      <=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_xa_2,V_z_2)
          & c_Orderings_Oord__class_Oless__eq(T_a,V_ya_2,V_z_2) ) ) ) )).

fof(fact_le__iff__sup,axiom,(
    ! [V_ya_2,V_xa_2,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => ( c_Orderings_Oord__class_Oless__eq(T_a,V_xa_2,V_ya_2)
      <=> c_Lattices_Osemilattice__sup__class_Osup(T_a,V_xa_2,V_ya_2) = V_ya_2 ) ) )).

fof(fact_sup__ge2,axiom,(
    ! [V_x,V_y,T_a] :
      ( class_Lattices_Osemilattice__sup(T_a)
     => c_Orderings_Oord__class_Oless__eq(T_a,V_y,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)
     => c_Orderings_Oord__class_Oless__eq(T_a,V_y,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)
     => c_Orderings_Oord__class_Oless__eq(T_a,V_x,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)
     => c_Orderings_Oord__class_Oless__eq(T_a,V_x,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_y)) ) )).

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

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

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

fof(fact_Un__mono,axiom,(
    ! [V_D_2,V_B_2,V_C_2,V_A_2,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_C_2)
     => ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_D_2)
       => c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_C_2,V_D_2)) ) ) )).

fof(fact_Un__least,axiom,(
    ! [V_B_2,V_C_2,V_A_2,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_C_2)
     => ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2)
       => c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_C_2) ) ) )).

fof(fact_Un__absorb2,axiom,(
    ! [V_A_2,V_B_2,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_A_2)
     => c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) = V_A_2 ) )).

fof(fact_Un__absorb1,axiom,(
    ! [V_B_2,V_A_2,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)
     => c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) = V_B_2 ) )).

fof(fact_subset__Un__eq,axiom,(
    ! [V_B_2,V_A_2,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)
    <=> c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) = V_B_2 ) )).

fof(fact_Un__upper2,axiom,(
    ! [V_A_2,V_B_2,T_a] : c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)) )).

fof(fact_Un__upper1,axiom,(
    ! [V_B_2,V_A_2,T_a] : c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)) )).

fof(fact_double__diff,axiom,(
    ! [V_C_2,V_B_2,V_A_2,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)
     => ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2)
       => c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_B_2,c_Groups_Ominus__class_Ominus(tc_fun(T_a,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_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_C_2)
     => ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_D_2,V_B_2)
       => c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_C_2,V_D_2)) ) ) )).

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

fof(fact_acc__subset,axiom,(
    ! [V_R2_2,V_R1_2,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_R1_2,V_R2_2)
     => c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Wellfounded_Oacc(T_a,V_R2_2),c_Wellfounded_Oacc(T_a,V_R1_2)) ) )).

fof(fact_rev__predicate1D,axiom,(
    ! [V_Q_2,T_a,V_xa_2,V_P_2] :
      ( hBOOL(hAPP(V_P_2,V_xa_2))
     => ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_P_2,V_Q_2)
       => hBOOL(hAPP(V_Q_2,V_xa_2)) ) ) )).

fof(fact_predicate1D,axiom,(
    ! [V_xa_2,V_Q_2,V_P_2,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_P_2,V_Q_2)
     => ( hBOOL(hAPP(V_P_2,V_xa_2))
       => hBOOL(hAPP(V_Q_2,V_xa_2)) ) ) )).

fof(fact_subset__refl,axiom,(
    ! [V_A_2,T_a] : c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_A_2) )).

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

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

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

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

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

fof(fact_Domain__mono,axiom,(
    ! [V_s_2,V_r_2,T_b,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),V_r_2,V_s_2)
     => c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Relation_ODomain(T_a,T_b,V_r_2),c_Relation_ODomain(T_a,T_b,V_s_2)) ) )).

fof(fact_acyclic__subset,axiom,(
    ! [V_r_2,V_s_2,T_a] :
      ( c_Wellfounded_Oacyclic(T_a,V_s_2)
     => ( c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,V_s_2)
       => c_Wellfounded_Oacyclic(T_a,V_r_2) ) ) )).

fof(fact_antisym__subset,axiom,(
    ! [V_s_2,V_r_2,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,V_s_2)
     => ( c_Relation_Oantisym(T_a,V_s_2)
       => c_Relation_Oantisym(T_a,V_r_2) ) ) )).

fof(fact_rtrancl__mono,axiom,(
    ! [V_s_2,V_r_2,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,V_s_2)
     => c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),c_Transitive__Closure_Ortrancl(T_a,V_r_2),c_Transitive__Closure_Ortrancl(T_a,V_s_2)) ) )).

fof(fact_rtrancl__subset,axiom,(
    ! [V_S_2,V_R_2,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_R_2,V_S_2)
     => ( c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_S_2,c_Transitive__Closure_Ortrancl(T_a,V_R_2))
       => c_Transitive__Closure_Ortrancl(T_a,V_S_2) = c_Transitive__Closure_Ortrancl(T_a,V_R_2) ) ) )).

fof(fact_rtrancl__subset__rtrancl,axiom,(
    ! [V_s_2,V_r_2,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,c_Transitive__Closure_Ortrancl(T_a,V_s_2))
     => c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),c_Transitive__Closure_Ortrancl(T_a,V_r_2),c_Transitive__Closure_Ortrancl(T_a,V_s_2)) ) )).

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

fof(fact_insert__subset,axiom,(
    ! [V_B_2,V_A_2,V_xa_2,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Set_Oinsert(T_a,V_xa_2,V_A_2),V_B_2)
    <=> ( c_member(T_a,V_xa_2,V_B_2)
        & c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) ) ) )).

fof(fact_subset__insert,axiom,(
    ! [V_B_2,V_A_2,V_xa_2,T_a] :
      ( ~ c_member(T_a,V_xa_2,V_A_2)
     => ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Set_Oinsert(T_a,V_xa_2,V_B_2))
      <=> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) ) ) )).

fof(fact_subset__singletonD,axiom,(
    ! [V_xa_2,V_A_2,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Set_Oinsert(T_a,V_xa_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))))
     => ( V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))
        | V_A_2 = c_Set_Oinsert(T_a,V_xa_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool))) ) ) )).

fof(fact_Diff__subset__conv,axiom,(
    ! [V_C_2,V_B_2,V_A_2,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_C_2)
    <=> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2)) ) )).

fof(fact_Diff__partition,axiom,(
    ! [V_B_2,V_A_2,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)
     => c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_A_2)) = V_B_2 ) )).

fof(fact_trancl__mono,axiom,(
    ! [V_s_2,V_r_2,V_p_2,T_a] :
      ( c_member(tc_prod(T_a,T_a),V_p_2,c_Transitive__Closure_Otrancl(T_a,V_r_2))
     => ( c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_r_2,V_s_2)
       => c_member(tc_prod(T_a,T_a),V_p_2,c_Transitive__Closure_Otrancl(T_a,V_s_2)) ) ) )).

fof(fact_Image__closed__trancl,axiom,(
    ! [V_X_2,V_r_2,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Relation_OImage(T_a,T_a,V_r_2,V_X_2),V_X_2)
     => c_Relation_OImage(T_a,T_a,c_Transitive__Closure_Ortrancl(T_a,V_r_2),V_X_2) = V_X_2 ) )).

fof(fact_rtrancl__Un__subset,axiom,(
    ! [V_S_2,V_R_2,T_a] : c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),c_Transitive__Closure_Ortrancl(T_a,V_R_2),c_Transitive__Closure_Ortrancl(T_a,V_S_2)),c_Transitive__Closure_Ortrancl(T_a,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),V_R_2,V_S_2))) )).

fof(fact_wfE__pf,axiom,(
    ! [V_A_2,V_R_2,T_a] :
      ( c_Wellfounded_Owf(T_a,V_R_2)
     => ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Relation_OImage(T_a,T_a,V_R_2,V_A_2))
       => V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)) ) ) )).

fof(fact_Domain__Diff__subset,axiom,(
    ! [V_B_2,V_A_2,T_b,T_a] : c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),c_Relation_ODomain(T_a,T_b,V_A_2),c_Relation_ODomain(T_a,T_b,V_B_2)),c_Relation_ODomain(T_a,T_b,c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(T_a,T_b),tc_HOL_Obool),V_A_2,V_B_2))) )).

fof(fact_Range__Diff__subset,axiom,(
    ! [V_B_2,V_A_2,T_b,T_a] : c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),c_Relation_ORange(T_b,T_a,V_A_2),c_Relation_ORange(T_b,T_a,V_B_2)),c_Relation_ORange(T_b,T_a,c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(T_b,T_a),tc_HOL_Obool),V_A_2,V_B_2))) )).

fof(fact_less__than__def,axiom,(
    c_Wellfounded_Oless__than = c_Transitive__Closure_Otrancl(tc_Nat_Onat,c_Wellfounded_Opred__nat) )).

fof(fact_subset__insert__iff,axiom,(
    ! [V_B_2,V_xa_2,V_A_2,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Set_Oinsert(T_a,V_xa_2,V_B_2))
    <=> ( ( c_member(T_a,V_xa_2,V_A_2)
         => c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Set_Oinsert(T_a,V_xa_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))),V_B_2) )
        & ( ~ c_member(T_a,V_xa_2,V_A_2)
         => c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) ) ) ) )).

fof(fact_diff__single__insert,axiom,(
    ! [V_B_2,V_xa_2,V_A_2,T_a] :
      ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Set_Oinsert(T_a,V_xa_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))),V_B_2)
     => ( c_member(T_a,V_xa_2,V_A_2)
       => c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Set_Oinsert(T_a,V_xa_2,V_B_2)) ) ) )).

fof(fact_wf__pred__nat,axiom,(
    c_Wellfounded_Owf(tc_Nat_Onat,c_Wellfounded_Opred__nat) )).

fof(fact_psubset__insert__iff,axiom,(
    ! [V_B_2,V_xa_2,V_A_2,T_a] :
      ( c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Set_Oinsert(T_a,V_xa_2,V_B_2))
    <=> ( ( c_member(T_a,V_xa_2,V_B_2)
         => c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) )
        & ( ~ c_member(T_a,V_xa_2,V_B_2)
         => ( ( c_member(T_a,V_xa_2,V_A_2)
             => c_Orderings_Oord__class_Oless(tc_fun(T_a,tc_HOL_Obool),c_Groups_Ominus__class_Ominus(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Set_Oinsert(T_a,V_xa_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))),V_B_2) )
            & ( ~ c_member(T_a,V_xa_2,V_A_2)
             => c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2) ) ) ) ) ) )).

fof(fact_subset__Image__Image__iff,axiom,(
    ! [V_B_2,V_A_2,V_r_2,T_a] :
      ( c_Order__Relation_Opreorder__on(T_a,c_Relation_OField(T_a,V_r_2),V_r_2)
     => ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Relation_OField(T_a,V_r_2))
       => ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_B_2,c_Relation_OField(T_a,V_r_2))
         => ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Relation_OImage(T_a,T_a,V_r_2,V_A_2),c_Relation_OImage(T_a,T_a,V_r_2,V_B_2))
          <=> ! [B_x] :
                ( c_member(T_a,B_x,V_A_2)
               => ? [B_xa] :
                    ( c_member(T_a,B_xa,V_B_2)
                    & c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,B_xa,B_x),V_r_2) ) ) ) ) ) ) )).

fof(fact_ImageE,axiom,(
    ! [V_A_2,V_r_2,T_b,V_b_2,T_a] :
      ( c_member(T_a,V_b_2,c_Relation_OImage(T_b,T_a,V_r_2,V_A_2))
     => ~ ! [B_x] :
            ( c_member(tc_prod(T_b,T_a),c_Product__Type_OPair(T_b,T_a,B_x,V_b_2),V_r_2)
           => ~ c_member(T_b,B_x,V_A_2) ) ) )).

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

fof(fact_wfP__subset,axiom,(
    ! [V_p_2,V_r_2,T_a] :
      ( c_Wellfounded_OwfP(T_a,V_r_2)
     => ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_fun(T_a,tc_HOL_Obool)),V_p_2,V_r_2)
       => c_Wellfounded_OwfP(T_a,V_p_2) ) ) )).

fof(fact_mlex__leq,axiom,(
    ! [V_R_2,T_a,V_ya_2,V_xa_2,V_f_2] :
      ( c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,hAPP(V_f_2,V_xa_2),hAPP(V_f_2,V_ya_2))
     => ( c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_ya_2),V_R_2)
       => c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_ya_2),c_Wellfounded_Omlex__prod(T_a,V_f_2,V_R_2)) ) ) )).

fof(fact_pred__nat__trancl__eq__le,axiom,(
    ! [V_n_2,V_m_2] :
      ( c_member(tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat,V_m_2,V_n_2),c_Transitive__Closure_Ortrancl(tc_Nat_Onat,c_Wellfounded_Opred__nat))
    <=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat,V_m_2,V_n_2) ) )).

fof(fact_irrefl__tranclI,axiom,(
    ! [V_xa_2,V_r_2,T_a] :
      ( c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool),c_Relation_Oconverse(T_a,T_a,V_r_2),c_Transitive__Closure_Ortrancl(T_a,V_r_2)) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_a,T_a),tc_HOL_Obool))
     => ~ c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_xa_2,V_xa_2),c_Transitive__Closure_Otrancl(T_a,V_r_2)) ) )).

fof(fact_subset__equiv__class,axiom,(
    ! [V_a_2,V_b_2,V_r_2,V_A_2,T_a] :
      ( c_Equiv__Relations_Oequiv(T_a,V_A_2,V_r_2)
     => ( c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Relation_OImage(T_a,T_a,V_r_2,c_Set_Oinsert(T_a,V_b_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))),c_Relation_OImage(T_a,T_a,V_r_2,c_Set_Oinsert(T_a,V_a_2,c_Orderings_Obot__class_Obot(tc_fun(T_a,tc_HOL_Obool)))))
       => ( c_member(T_a,V_b_2,V_A_2)
         => c_member(tc_prod(T_a,T_a),c_Product__Type_OPair(T_a,T_a,V_a_2,V_b_2),V_r_2) ) ) ) )).

fof(fact_inf1I,axiom,(
    ! [T_a,V_B_2,V_xa_2,V_A_2] :
      ( hBOOL(hAPP(V_A_2,V_xa_2))
     => ( hBOOL(hAPP(V_B_2,V_xa_2))
       => hBOOL(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_xa_2)) ) ) )).

fof(fact_inf1E,axiom,(
    ! [V_xa_2,V_B_2,V_A_2,T_a] :
      ( hBOOL(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_xa_2))
     => ~ ( hBOOL(hAPP(V_A_2,V_xa_2))
         => ~ hBOOL(hAPP(V_B_2,V_xa_2)) ) ) )).

fof(fact_IntI,axiom,(
    ! [V_B_2,V_A_2,V_c_2,T_a] :
      ( c_member(T_a,V_c_2,V_A_2)
     => ( c_member(T_a,V_c_2,V_B_2)
       => c_member(T_a,V_c_2,c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2)) ) ) )).

fof(fact_IntE,axiom,(
    ! [V_B_2,V_A_2,V_c_2,T_a] :
      ( c_member(T_a,V_c_2,c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2))
     => ~ ( c_member(T_a,V_c_2,V_A_2)
         => ~ c_member(T_a,V_c_2,V_B_2) ) ) )).

fof(fact_Image__Int__subset,axiom,(
    ! [V_B_2,V_A_2,V_R_2,T_b,T_a] : c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),c_Relation_OImage(T_b,T_a,V_R_2,c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool),V_A_2,V_B_2)),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),c_Relation_OImage(T_b,T_a,V_R_2,V_A_2),c_Relation_OImage(T_b,T_a,V_R_2,V_B_2))) )).

fof(fact_Un__Int__assoc__eq,axiom,(
    ! [V_C_2,V_B_2,V_A_2,T_a] :
      ( c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,V_B_2),V_C_2) = c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_a,tc_HOL_Obool),V_A_2,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_a,tc_HOL_Obool),V_B_2,V_C_2))
    <=> c_Orderings_Oord__class_Oless__eq(tc_fun(T_a,tc_HOL_Obool),V_C_2,V_A_2) ) )).

fof(fact_distrib__inf__le,axiom,(
    ! [V_z,V_y,V_x,T_a] :
      ( class_Lattices_Olattice(T_a)
     => c_Orderings_Oord__class_Oless__eq(T_a,c_Lattices_Osemilattice__sup__class_Osup(T_a,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x,V_y),c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x,V_z)),c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_y,V_z))) ) )).

fof(fact_distrib__sup__le,axiom,(
    ! [V_z,V_y,V_x,T_a] :
      ( class_Lattices_Olattice(T_a)
     => c_Orderings_Oord__class_Oless__eq(T_a,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_y,V_z)),c_Lattices_Osemilattice__inf__class_Oinf(T_a,c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_y),c_Lattices_Osemilattice__sup__class_Osup(T_a,V_x,V_z))) ) )).

fof(fact_inf__sup__ord_I1_J,axiom,(
    ! [V_y,V_x,T_a] :
      ( class_Lattices_Olattice(T_a)
     => c_Orderings_Oord__class_Oless__eq(T_a,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)
     => c_Orderings_Oord__class_Oless__eq(T_a,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)
     => c_Orderings_Oord__class_Oless__eq(T_a,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)
     => c_Orderings_Oord__class_Oless__eq(T_a,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_x,V_y),V_y) ) )).

fof(fact_le__iff__inf,axiom,(
    ! [V_ya_2,V_xa_2,T_a] :
      ( class_Lattices_Osemilattice__inf(T_a)
     => ( c_Orderings_Oord__class_Oless__eq(T_a,V_xa_2,V_ya_2)
      <=> c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_xa_2,V_ya_2) = V_xa_2 ) ) )).

fof(fact_le__inf__iff,axiom,(
    ! [V_z_2,V_ya_2,V_xa_2,T_a] :
      ( class_Lattices_Osemilattice__inf(T_a)
     => ( c_Orderings_Oord__class_Oless__eq(T_a,V_xa_2,c_Lattices_Osemilattice__inf__class_Oinf(T_a,V_ya_2,V_z_2))
      <=> ( c_Orderings_Oord__class_Oless__eq(T_a,V_xa_2,V_ya_2)
          & c_Orderings_Oord__class_Oless__eq(T_a,V_xa_2,V_z_2) ) ) ) )).

%----Arity declarations (29)
fof(arity_HOL__Obool__Lattices_Obounded__lattice,axiom,(
    class_Lattices_Obounded__lattice(tc_HOL_Obool) )).

fof(arity_fun__Lattices_Obounded__lattice,axiom,(
    ! [T_2,T_1] :
      ( class_Lattices_Obounded__lattice(T_1)
     => class_Lattices_Obounded__lattice(tc_fun(T_2,T_1)) ) )).

fof(arity_fun__Lattices_Obounded__lattice__bot,axiom,(
    ! [T_2,T_1] :
      ( class_Lattices_Obounded__lattice(T_1)
     => class_Lattices_Obounded__lattice__bot(tc_fun(T_2,T_1)) ) )).

fof(arity_fun__Lattices_Osemilattice__sup,axiom,(
    ! [T_2,T_1] :
      ( class_Lattices_Olattice(T_1)
     => class_Lattices_Osemilattice__sup(tc_fun(T_2,T_1)) ) )).

fof(arity_fun__Lattices_Osemilattice__inf,axiom,(
    ! [T_2,T_1] :
      ( class_Lattices_Olattice(T_1)
     => class_Lattices_Osemilattice__inf(tc_fun(T_2,T_1)) ) )).

fof(arity_fun__Orderings_Opreorder,axiom,(
    ! [T_2,T_1] :
      ( class_Orderings_Opreorder(T_1)
     => class_Orderings_Opreorder(tc_fun(T_2,T_1)) ) )).

fof(arity_fun__Lattices_Olattice,axiom,(
    ! [T_2,T_1] :
      ( class_Lattices_Olattice(T_1)
     => class_Lattices_Olattice(tc_fun(T_2,T_1)) ) )).

fof(arity_fun__Orderings_Oorder,axiom,(
    ! [T_2,T_1] :
      ( class_Orderings_Oorder(T_1)
     => class_Orderings_Oorder(tc_fun(T_2,T_1)) ) )).

fof(arity_fun__Orderings_Oord,axiom,(
    ! [T_2,T_1] :
      ( class_Orderings_Oord(T_1)
     => class_Orderings_Oord(tc_fun(T_2,T_1)) ) )).

fof(arity_fun__Orderings_Obot,axiom,(
    ! [T_2,T_1] :
      ( class_Orderings_Obot(T_1)
     => class_Orderings_Obot(tc_fun(T_2,T_1)) ) )).

fof(arity_fun__Groups_Ominus,axiom,(
    ! [T_2,T_1] :
      ( class_Groups_Ominus(T_1)
     => class_Groups_Ominus(tc_fun(T_2,T_1)) ) )).

fof(arity_Nat__Onat__Lattices_Osemilattice__sup,axiom,(
    class_Lattices_Osemilattice__sup(tc_Nat_Onat) )).

fof(arity_Nat__Onat__Lattices_Osemilattice__inf,axiom,(
    class_Lattices_Osemilattice__inf(tc_Nat_Onat) )).

fof(arity_Nat__Onat__Orderings_Opreorder,axiom,(
    class_Orderings_Opreorder(tc_Nat_Onat) )).

fof(arity_Nat__Onat__Orderings_Olinorder,axiom,(
    class_Orderings_Olinorder(tc_Nat_Onat) )).

fof(arity_Nat__Onat__Lattices_Olattice,axiom,(
    class_Lattices_Olattice(tc_Nat_Onat) )).

fof(arity_Nat__Onat__Orderings_Oorder,axiom,(
    class_Orderings_Oorder(tc_Nat_Onat) )).

fof(arity_Nat__Onat__Orderings_Oord,axiom,(
    class_Orderings_Oord(tc_Nat_Onat) )).

fof(arity_Nat__Onat__Orderings_Obot,axiom,(
    class_Orderings_Obot(tc_Nat_Onat) )).

fof(arity_Nat__Onat__Groups_Ominus,axiom,(
    class_Groups_Ominus(tc_Nat_Onat) )).

fof(arity_HOL__Obool__Lattices_Obounded__lattice__bot,axiom,(
    class_Lattices_Obounded__lattice__bot(tc_HOL_Obool) )).

fof(arity_HOL__Obool__Lattices_Osemilattice__sup,axiom,(
    class_Lattices_Osemilattice__sup(tc_HOL_Obool) )).

fof(arity_HOL__Obool__Lattices_Osemilattice__inf,axiom,(
    class_Lattices_Osemilattice__inf(tc_HOL_Obool) )).

fof(arity_HOL__Obool__Orderings_Opreorder,axiom,(
    class_Orderings_Opreorder(tc_HOL_Obool) )).

fof(arity_HOL__Obool__Lattices_Olattice,axiom,(
    class_Lattices_Olattice(tc_HOL_Obool) )).

fof(arity_HOL__Obool__Orderings_Oorder,axiom,(
    class_Orderings_Oorder(tc_HOL_Obool) )).

fof(arity_HOL__Obool__Orderings_Oord,axiom,(
    class_Orderings_Oord(tc_HOL_Obool) )).

fof(arity_HOL__Obool__Orderings_Obot,axiom,(
    class_Orderings_Obot(tc_HOL_Obool) )).

fof(arity_HOL__Obool__Groups_Ominus,axiom,(
    class_Groups_Ominus(tc_HOL_Obool) )).

%----Helper facts (2)
fof(help_c__fequal__1,axiom,(
    ! [V_y_2,V_x_2] :
      ( ~ hBOOL(hAPP(c_fequal(V_x_2),V_y_2))
      | V_x_2 = V_y_2 ) )).

fof(help_c__fequal__2,axiom,(
    ! [V_y_2,V_x_2] :
      ( V_x_2 != V_y_2
      | hBOOL(hAPP(c_fequal(V_x_2),V_y_2)) ) )).

%----Conjectures (3)
fof(conj_0,hypothesis,(
    v_x != v_y )).

fof(conj_1,hypothesis,
    ( ! [B_x,B_y] :
        ( c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,B_x,B_y),v_L)
       => ! [B_z] :
            ( c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,B_y,B_z),v_L)
           => c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,B_x,B_z),v_L) ) )
    & ! [B_x] : ~ c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,B_x,B_x),v_L)
    & ! [B_x,B_y] :
        ( B_x != B_y
       => ( c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,B_x,B_y),v_L)
          | c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,B_y,B_x),v_L) ) ) )).

fof(conj_2,conjecture,
    ( ! [B_x,B_y] :
        ( ( ( c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,B_x,B_y),v_L)
            & B_x != v_x
            & B_y != v_x )
         => ! [B_z] :
              ( ( ( c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,B_y,B_z),v_L)
                  & B_z != v_x )
               => c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,B_x,B_z),v_L) )
              & ( ( B_z = v_x
                  & c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,B_y,v_y),v_L) )
               => c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,B_x,v_y),v_L) ) ) )
        & ( ( B_y = v_x
            & c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,B_x,v_y),v_L)
            & B_x != v_x )
         => ! [B_z] :
              ( ( c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,v_y,B_z),v_L)
                & B_z != v_x )
             => c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,B_x,B_z),v_L) ) )
        & ( ( B_x = v_x
            & c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,v_y,B_y),v_L)
            & B_y != v_x )
         => ! [B_z] :
              ( ( ( c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,B_y,B_z),v_L)
                  & B_z != v_x )
               => ( B_z = v_y
                  | c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,v_y,B_z),v_L) ) )
              & ( B_z = v_x
               => ~ c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,B_y,v_y),v_L) ) ) ) )
    & ! [B_x,B_y] :
        ( B_x != B_y
       => ( ( B_x = v_x
            & B_y = v_y )
          | ( c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,B_x,B_y),v_L)
            & B_x != v_x
            & B_y != v_x )
          | ( B_y = v_x
            & c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,B_x,v_y),v_L)
            & B_x != v_x )
          | ( B_x = v_x
            & c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,v_y,B_y),v_L)
            & B_y != v_x )
          | ( B_y = v_x
            & B_x = v_y )
          | ( c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,B_y,B_x),v_L)
            & B_y != v_x
            & B_x != v_x )
          | ( B_x = v_x
            & c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,B_y,v_y),v_L)
            & B_y != v_x )
          | ( B_y = v_x
            & c_member(tc_prod(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt),c_Product__Type_OPair(tc_Arrow__Order__Mirabelle_Oalt,tc_Arrow__Order__Mirabelle_Oalt,v_y,B_x),v_L)
            & B_x != v_x ) ) ) )).

%------------------------------------------------------------------------------