TPTP Problem File: NUM757^4.p

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%------------------------------------------------------------------------------
% File     : NUM757^4 : TPTP v7.4.0. Released v7.1.0.
% Domain   : Number theory
% Problem  : Grundlagen problem satz63c
% Version  : [Bro17] axioms : Especial.
% English  :

% Refs     : [Bro17] Brown (2017), Email to G. Sutcliffe
% Source   : [Br017]
% Names    :

% Status   : Theorem
% Rating   : 0.86 v7.4.0, 0.89 v7.2.0, 0.88 v7.1.0
% Syntax   : Number of formulae    :  594 (   0 unit; 177 type; 170 defn)
%            Number of atoms       : 5736 ( 200 equality;2441 variable)
%            Maximal formula depth :   22 (   8 average)
%            Number of connectives : 4926 (   7   ~;   4   |;  14   &;4644   @)
%                                         (   3 <=>; 254  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :  534 ( 534   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  180 ( 177   :;   0   =)
%            Number of variables   : 1572 (  32 sgn; 151   !;   8   ?;1413   ^)
%                                         (1572   :;   0  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      : TH0_THM_EQU_NAR

% Comments :
%------------------------------------------------------------------------------
include('Axioms/NUM007^0.ax').
%------------------------------------------------------------------------------
thf(satz1,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( ( nis @ X0 @ X1 )
           => ( nis @ ( ordsucc @ X0 ) @ ( ordsucc @ X1 ) ) ) ) )).

thf(typ_d_22_prop1,type,(
    d_22_prop1: $i > $o )).

thf(def_d_22_prop1,definition,
    ( d_22_prop1
    = ( ^ [X0: $i] :
          ( nis @ ( ordsucc @ X0 ) @ X0 ) ) )).

thf(satz2,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( nis @ ( ordsucc @ X0 ) @ X0 ) )).

thf(typ_d_23_prop1,type,(
    d_23_prop1: $i > $o )).

thf(def_d_23_prop1,definition,
    ( d_23_prop1
    = ( ^ [X0: $i] :
          ( l_or @ ( n_is @ X0 @ n_1 )
          @ ( n_some
            @ ^ [X1: $i] :
                ( n_is @ X0 @ ( ordsucc @ X1 ) ) ) ) ) )).

thf(satz3,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( ( nis @ X0 @ n_1 )
       => ( n_some
          @ ^ [X1: $i] :
              ( n_is @ X0 @ ( ordsucc @ X1 ) ) ) ) )).

thf(satz3a,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( ( nis @ X0 @ n_1 )
       => ( n_one
          @ ^ [X1: $i] :
              ( n_is @ X0 @ ( ordsucc @ X1 ) ) ) ) )).

thf(typ_d_24_prop1,type,(
    d_24_prop1: $i > $o )).

thf(def_d_24_prop1,definition,
    ( d_24_prop1
    = ( ^ [X0: $i] :
          ( n_all
          @ ^ [X1: $i] :
              ( n_is @ ( ap @ X0 @ ( ordsucc @ X1 ) ) @ ( ordsucc @ ( ap @ X0 @ X1 ) ) ) ) ) )).

thf(typ_d_24_prop2,type,(
    d_24_prop2: $i > $i > $o )).

thf(def_d_24_prop2,definition,
    ( d_24_prop2
    = ( ^ [X0: $i,X1: $i] :
          ( d_and @ ( n_is @ ( ap @ X1 @ n_1 ) @ ( ordsucc @ X0 ) ) @ ( d_24_prop1 @ X1 ) ) ) )).

thf(typ_prop3,type,(
    prop3: $i > $i > $i > $o )).

thf(def_prop3,definition,
    ( prop3
    = ( ^ [X0: $i,X1: $i,X2: $i] :
          ( n_is @ ( ap @ X0 @ X2 ) @ ( ap @ X1 @ X2 ) ) ) )).

thf(typ_prop4,type,(
    prop4: $i > $o )).

thf(def_prop4,definition,
    ( prop4
    = ( ^ [X0: $i] :
          ( l_some
          @ ( d_Pi @ nat
            @ ^ [X1: $i] : nat )
          @ ( d_24_prop2 @ X0 ) ) ) )).

thf(typ_d_24_g,type,(
    d_24_g: $i > $i )).

thf(def_d_24_g,definition,
    ( d_24_g
    = ( ^ [X0: $i] :
          ( d_Sigma @ nat
          @ ^ [X1: $i] :
              ( ordsucc @ ( ap @ X0 @ X1 ) ) ) ) )).

thf(satz4,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( one
        @ ( d_Pi @ nat
          @ ^ [X1: $i] : nat )
        @ ^ [X1: $i] :
            ( d_and @ ( n_is @ ( ap @ X1 @ n_1 ) @ ( ordsucc @ X0 ) )
            @ ( n_all
              @ ^ [X2: $i] :
                  ( n_is @ ( ap @ X1 @ ( ordsucc @ X2 ) ) @ ( ordsucc @ ( ap @ X1 @ X2 ) ) ) ) ) ) )).

thf(typ_plus,type,(
    plus: $i > $i )).

thf(def_plus,definition,
    ( plus
    = ( ^ [X0: $i] :
          ( ind
          @ ( d_Pi @ nat
            @ ^ [X1: $i] : nat )
          @ ( d_24_prop2 @ X0 ) ) ) )).

thf(typ_n_pl,type,(
    n_pl: $i > $i > $i )).

thf(def_n_pl,definition,
    ( n_pl
    = ( ^ [X0: $i] :
          ( ap @ ( plus @ X0 ) ) ) )).

thf(satz4a,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( n_is @ ( n_pl @ X0 @ n_1 ) @ ( ordsucc @ X0 ) ) )).

thf(satz4b,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( n_is @ ( n_pl @ X0 @ ( ordsucc @ X1 ) ) @ ( ordsucc @ ( n_pl @ X0 @ X1 ) ) ) ) )).

thf(satz4c,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( n_is @ ( n_pl @ n_1 @ X0 ) @ ( ordsucc @ X0 ) ) )).

thf(satz4d,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( n_is @ ( n_pl @ ( ordsucc @ X0 ) @ X1 ) @ ( ordsucc @ ( n_pl @ X0 @ X1 ) ) ) ) )).

thf(satz4e,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( n_is @ ( ordsucc @ X0 ) @ ( n_pl @ X0 @ n_1 ) ) )).

thf(satz4f,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( n_is @ ( ordsucc @ ( n_pl @ X0 @ X1 ) ) @ ( n_pl @ X0 @ ( ordsucc @ X1 ) ) ) ) )).

thf(satz4g,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( n_is @ ( ordsucc @ X0 ) @ ( n_pl @ n_1 @ X0 ) ) )).

thf(satz4h,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( n_is @ ( ordsucc @ ( n_pl @ X0 @ X1 ) ) @ ( n_pl @ ( ordsucc @ X0 ) @ X1 ) ) ) )).

thf(typ_d_25_prop1,type,(
    d_25_prop1: $i > $i > $i > $o )).

thf(def_d_25_prop1,definition,
    ( d_25_prop1
    = ( ^ [X0: $i,X1: $i,X2: $i] :
          ( n_is @ ( n_pl @ ( n_pl @ X0 @ X1 ) @ X2 ) @ ( n_pl @ X0 @ ( n_pl @ X1 @ X2 ) ) ) ) )).

thf(satz5,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( n_is @ ( n_pl @ ( n_pl @ X0 @ X1 ) @ X2 ) @ ( n_pl @ X0 @ ( n_pl @ X1 @ X2 ) ) ) ) ) )).

thf(typ_d_26_prop1,type,(
    d_26_prop1: $i > $i > $o )).

thf(def_d_26_prop1,definition,
    ( d_26_prop1
    = ( ^ [X0: $i,X1: $i] :
          ( n_is @ ( n_pl @ X0 @ X1 ) @ ( n_pl @ X1 @ X0 ) ) ) )).

thf(satz6,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( n_is @ ( n_pl @ X0 @ X1 ) @ ( n_pl @ X1 @ X0 ) ) ) )).

thf(typ_d_27_prop1,type,(
    d_27_prop1: $i > $i > $o )).

thf(def_d_27_prop1,definition,
    ( d_27_prop1
    = ( ^ [X0: $i,X1: $i] :
          ( nis @ X1 @ ( n_pl @ X0 @ X1 ) ) ) )).

thf(satz7,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( nis @ X1 @ ( n_pl @ X0 @ X1 ) ) ) )).

thf(typ_d_28_prop1,type,(
    d_28_prop1: $i > $i > $i > $o )).

thf(def_d_28_prop1,definition,
    ( d_28_prop1
    = ( ^ [X0: $i,X1: $i,X2: $i] :
          ( nis @ ( n_pl @ X0 @ X1 ) @ ( n_pl @ X0 @ X2 ) ) ) )).

thf(satz8,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( ( nis @ X1 @ X2 )
               => ( nis @ ( n_pl @ X0 @ X1 ) @ ( n_pl @ X0 @ X2 ) ) ) ) ) )).

thf(satz8a,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( ( n_is @ ( n_pl @ X0 @ X1 ) @ ( n_pl @ X0 @ X2 ) )
               => ( n_is @ X1 @ X2 ) ) ) ) )).

thf(typ_diffprop,type,(
    diffprop: $i > $i > $i > $o )).

thf(def_diffprop,definition,
    ( diffprop
    = ( ^ [X0: $i,X1: $i,X2: $i] :
          ( n_is @ X0 @ ( n_pl @ X1 @ X2 ) ) ) )).

thf(satz8b,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( amone @ nat
            @ ^ [X2: $i] :
                ( n_is @ X0 @ ( n_pl @ X1 @ X2 ) ) ) ) )).

thf(typ_d_29_ii,type,(
    d_29_ii: $i > $i > $o )).

thf(def_d_29_ii,definition,
    ( d_29_ii
    = ( ^ [X0: $i,X1: $i] :
          ( n_some @ ( diffprop @ X0 @ X1 ) ) ) )).

thf(typ_iii,type,(
    iii: $i > $i > $o )).

thf(def_iii,definition,
    ( iii
    = ( ^ [X0: $i,X1: $i] :
          ( n_some @ ( diffprop @ X1 @ X0 ) ) ) )).

thf(typ_d_29_prop1,type,(
    d_29_prop1: $i > $i > $o )).

thf(def_d_29_prop1,definition,
    ( d_29_prop1
    = ( ^ [X0: $i,X1: $i] :
          ( or3 @ ( n_is @ X0 @ X1 ) @ ( d_29_ii @ X0 @ X1 ) @ ( iii @ X0 @ X1 ) ) ) )).

thf(satz9,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( orec3 @ ( n_is @ X0 @ X1 )
            @ ( n_some
              @ ^ [X2: $i] :
                  ( n_is @ X0 @ ( n_pl @ X1 @ X2 ) ) )
            @ ( n_some
              @ ^ [X2: $i] :
                  ( n_is @ X1 @ ( n_pl @ X0 @ X2 ) ) ) ) ) )).

thf(satz9a,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( or3 @ ( n_is @ X0 @ X1 ) @ ( n_some @ ( diffprop @ X0 @ X1 ) ) @ ( n_some @ ( diffprop @ X1 @ X0 ) ) ) ) )).

thf(satz9b,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( ec3 @ ( n_is @ X0 @ X1 ) @ ( n_some @ ( diffprop @ X0 @ X1 ) ) @ ( n_some @ ( diffprop @ X1 @ X0 ) ) ) ) )).

thf(satz10,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( orec3 @ ( n_is @ X0 @ X1 ) @ ( d_29_ii @ X0 @ X1 ) @ ( iii @ X0 @ X1 ) ) ) )).

thf(satz10a,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( or3 @ ( n_is @ X0 @ X1 ) @ ( d_29_ii @ X0 @ X1 ) @ ( iii @ X0 @ X1 ) ) ) )).

thf(satz10b,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( ec3 @ ( n_is @ X0 @ X1 ) @ ( d_29_ii @ X0 @ X1 ) @ ( iii @ X0 @ X1 ) ) ) )).

thf(satz11,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( ( d_29_ii @ X0 @ X1 )
           => ( iii @ X1 @ X0 ) ) ) )).

thf(satz12,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( ( iii @ X0 @ X1 )
           => ( d_29_ii @ X1 @ X0 ) ) ) )).

thf(typ_moreis,type,(
    moreis: $i > $i > $o )).

thf(def_moreis,definition,
    ( moreis
    = ( ^ [X0: $i,X1: $i] :
          ( l_or @ ( d_29_ii @ X0 @ X1 ) @ ( n_is @ X0 @ X1 ) ) ) )).

thf(typ_lessis,type,(
    lessis: $i > $i > $o )).

thf(def_lessis,definition,
    ( lessis
    = ( ^ [X0: $i,X1: $i] :
          ( l_or @ ( iii @ X0 @ X1 ) @ ( n_is @ X0 @ X1 ) ) ) )).

thf(satz13,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( ( moreis @ X0 @ X1 )
           => ( lessis @ X1 @ X0 ) ) ) )).

thf(satz14,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( ( lessis @ X0 @ X1 )
           => ( moreis @ X1 @ X0 ) ) ) )).

thf(satz10c,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( ( moreis @ X0 @ X1 )
           => ( d_not @ ( iii @ X0 @ X1 ) ) ) ) )).

thf(satz10d,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( ( lessis @ X0 @ X1 )
           => ( d_not @ ( d_29_ii @ X0 @ X1 ) ) ) ) )).

thf(satz10e,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( ( d_not @ ( d_29_ii @ X0 @ X1 ) )
           => ( lessis @ X0 @ X1 ) ) ) )).

thf(satz10f,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( ( d_not @ ( iii @ X0 @ X1 ) )
           => ( moreis @ X0 @ X1 ) ) ) )).

thf(satz10g,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( ( d_29_ii @ X0 @ X1 )
           => ( d_not @ ( lessis @ X0 @ X1 ) ) ) ) )).

thf(satz10h,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( ( iii @ X0 @ X1 )
           => ( d_not @ ( moreis @ X0 @ X1 ) ) ) ) )).

thf(satz10j,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( ( d_not @ ( moreis @ X0 @ X1 ) )
           => ( iii @ X0 @ X1 ) ) ) )).

thf(satz10k,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( ( d_not @ ( lessis @ X0 @ X1 ) )
           => ( d_29_ii @ X0 @ X1 ) ) ) )).

thf(satz15,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( ( iii @ X0 @ X1 )
               => ( ( iii @ X1 @ X2 )
                 => ( iii @ X0 @ X2 ) ) ) ) ) )).

thf(satz16a,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( ( lessis @ X0 @ X1 )
               => ( ( iii @ X1 @ X2 )
                 => ( iii @ X0 @ X2 ) ) ) ) ) )).

thf(satz16b,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( ( iii @ X0 @ X1 )
               => ( ( lessis @ X1 @ X2 )
                 => ( iii @ X0 @ X2 ) ) ) ) ) )).

thf(satz16c,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( ( moreis @ X0 @ X1 )
               => ( ( d_29_ii @ X1 @ X2 )
                 => ( d_29_ii @ X0 @ X2 ) ) ) ) ) )).

thf(satz16d,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( ( d_29_ii @ X0 @ X1 )
               => ( ( moreis @ X1 @ X2 )
                 => ( d_29_ii @ X0 @ X2 ) ) ) ) ) )).

thf(satz17,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( ( lessis @ X0 @ X1 )
               => ( ( lessis @ X1 @ X2 )
                 => ( lessis @ X0 @ X2 ) ) ) ) ) )).

thf(satz18,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( d_29_ii @ ( n_pl @ X0 @ X1 ) @ X0 ) ) )).

thf(satz18a,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( iii @ X0 @ ( n_pl @ X0 @ X1 ) ) ) )).

thf(satz18b,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( d_29_ii @ ( ordsucc @ X0 ) @ X0 ) )).

thf(satz18c,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( iii @ X0 @ ( ordsucc @ X0 ) ) )).

thf(satz19a,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( ( d_29_ii @ X0 @ X1 )
               => ( d_29_ii @ ( n_pl @ X0 @ X2 ) @ ( n_pl @ X1 @ X2 ) ) ) ) ) )).

thf(satz19b,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( ( n_is @ X0 @ X1 )
               => ( n_is @ ( n_pl @ X0 @ X2 ) @ ( n_pl @ X1 @ X2 ) ) ) ) ) )).

thf(satz19c,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( ( iii @ X0 @ X1 )
               => ( iii @ ( n_pl @ X0 @ X2 ) @ ( n_pl @ X1 @ X2 ) ) ) ) ) )).

thf(satz19d,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( ( d_29_ii @ X0 @ X1 )
               => ( d_29_ii @ ( n_pl @ X2 @ X0 ) @ ( n_pl @ X2 @ X1 ) ) ) ) ) )).

thf(satz19e,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( ( n_is @ X0 @ X1 )
               => ( n_is @ ( n_pl @ X2 @ X0 ) @ ( n_pl @ X2 @ X1 ) ) ) ) ) )).

thf(satz19f,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( ( iii @ X0 @ X1 )
               => ( iii @ ( n_pl @ X2 @ X0 ) @ ( n_pl @ X2 @ X1 ) ) ) ) ) )).

thf(satz19g,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ nat )
                @ ^ [X3: $i] :
                    ( ( n_is @ X0 @ X1 )
                   => ( ( d_29_ii @ X2 @ X3 )
                     => ( d_29_ii @ ( n_pl @ X0 @ X2 ) @ ( n_pl @ X1 @ X3 ) ) ) ) ) ) ) )).

thf(satz19h,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ nat )
                @ ^ [X3: $i] :
                    ( ( n_is @ X0 @ X1 )
                   => ( ( d_29_ii @ X2 @ X3 )
                     => ( d_29_ii @ ( n_pl @ X2 @ X0 ) @ ( n_pl @ X3 @ X1 ) ) ) ) ) ) ) )).

thf(satz19j,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ nat )
                @ ^ [X3: $i] :
                    ( ( n_is @ X0 @ X1 )
                   => ( ( iii @ X2 @ X3 )
                     => ( iii @ ( n_pl @ X0 @ X2 ) @ ( n_pl @ X1 @ X3 ) ) ) ) ) ) ) )).

thf(satz19k,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ nat )
                @ ^ [X3: $i] :
                    ( ( n_is @ X0 @ X1 )
                   => ( ( iii @ X2 @ X3 )
                     => ( iii @ ( n_pl @ X2 @ X0 ) @ ( n_pl @ X3 @ X1 ) ) ) ) ) ) ) )).

thf(satz19l,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( ( moreis @ X0 @ X1 )
               => ( moreis @ ( n_pl @ X0 @ X2 ) @ ( n_pl @ X1 @ X2 ) ) ) ) ) )).

thf(satz19m,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( ( moreis @ X0 @ X1 )
               => ( moreis @ ( n_pl @ X2 @ X0 ) @ ( n_pl @ X2 @ X1 ) ) ) ) ) )).

thf(satz19n,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( ( lessis @ X0 @ X1 )
               => ( lessis @ ( n_pl @ X0 @ X2 ) @ ( n_pl @ X1 @ X2 ) ) ) ) ) )).

thf(satz19o,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( ( lessis @ X0 @ X1 )
               => ( lessis @ ( n_pl @ X2 @ X0 ) @ ( n_pl @ X2 @ X1 ) ) ) ) ) )).

thf(satz20a,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( ( d_29_ii @ ( n_pl @ X0 @ X2 ) @ ( n_pl @ X1 @ X2 ) )
               => ( d_29_ii @ X0 @ X1 ) ) ) ) )).

thf(satz20b,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( ( n_is @ ( n_pl @ X0 @ X2 ) @ ( n_pl @ X1 @ X2 ) )
               => ( n_is @ X0 @ X1 ) ) ) ) )).

thf(satz20c,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( ( iii @ ( n_pl @ X0 @ X2 ) @ ( n_pl @ X1 @ X2 ) )
               => ( iii @ X0 @ X1 ) ) ) ) )).

thf(satz20d,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( ( d_29_ii @ ( n_pl @ X2 @ X0 ) @ ( n_pl @ X2 @ X1 ) )
               => ( d_29_ii @ X0 @ X1 ) ) ) ) )).

thf(satz20e,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( ( n_is @ ( n_pl @ X2 @ X0 ) @ ( n_pl @ X2 @ X1 ) )
               => ( n_is @ X0 @ X1 ) ) ) ) )).

thf(satz20f,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( ( iii @ ( n_pl @ X2 @ X0 ) @ ( n_pl @ X2 @ X1 ) )
               => ( iii @ X0 @ X1 ) ) ) ) )).

thf(satz21,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ nat )
                @ ^ [X3: $i] :
                    ( ( d_29_ii @ X0 @ X1 )
                   => ( ( d_29_ii @ X2 @ X3 )
                     => ( d_29_ii @ ( n_pl @ X0 @ X2 ) @ ( n_pl @ X1 @ X3 ) ) ) ) ) ) ) )).

thf(satz21a,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ nat )
                @ ^ [X3: $i] :
                    ( ( iii @ X0 @ X1 )
                   => ( ( iii @ X2 @ X3 )
                     => ( iii @ ( n_pl @ X0 @ X2 ) @ ( n_pl @ X1 @ X3 ) ) ) ) ) ) ) )).

thf(satz22a,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ nat )
                @ ^ [X3: $i] :
                    ( ( moreis @ X0 @ X1 )
                   => ( ( d_29_ii @ X2 @ X3 )
                     => ( d_29_ii @ ( n_pl @ X0 @ X2 ) @ ( n_pl @ X1 @ X3 ) ) ) ) ) ) ) )).

thf(satz22b,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ nat )
                @ ^ [X3: $i] :
                    ( ( d_29_ii @ X0 @ X1 )
                   => ( ( moreis @ X2 @ X3 )
                     => ( d_29_ii @ ( n_pl @ X0 @ X2 ) @ ( n_pl @ X1 @ X3 ) ) ) ) ) ) ) )).

thf(satz22c,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ nat )
                @ ^ [X3: $i] :
                    ( ( lessis @ X0 @ X1 )
                   => ( ( iii @ X2 @ X3 )
                     => ( iii @ ( n_pl @ X0 @ X2 ) @ ( n_pl @ X1 @ X3 ) ) ) ) ) ) ) )).

thf(satz22d,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ nat )
                @ ^ [X3: $i] :
                    ( ( iii @ X0 @ X1 )
                   => ( ( lessis @ X2 @ X3 )
                     => ( iii @ ( n_pl @ X0 @ X2 ) @ ( n_pl @ X1 @ X3 ) ) ) ) ) ) ) )).

thf(satz23,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ nat )
                @ ^ [X3: $i] :
                    ( ( moreis @ X0 @ X1 )
                   => ( ( moreis @ X2 @ X3 )
                     => ( moreis @ ( n_pl @ X0 @ X2 ) @ ( n_pl @ X1 @ X3 ) ) ) ) ) ) ) )).

thf(satz23a,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ nat )
                @ ^ [X3: $i] :
                    ( ( lessis @ X0 @ X1 )
                   => ( ( lessis @ X2 @ X3 )
                     => ( lessis @ ( n_pl @ X0 @ X2 ) @ ( n_pl @ X1 @ X3 ) ) ) ) ) ) ) )).

thf(satz24,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( moreis @ X0 @ n_1 ) )).

thf(satz24a,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ( lessis @ n_1 ) )).

thf(satz24b,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( d_29_ii @ ( ordsucc @ X0 ) @ n_1 ) )).

thf(satz24c,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( iii @ n_1 @ ( ordsucc @ X0 ) ) )).

thf(satz25,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( ( d_29_ii @ X1 @ X0 )
           => ( moreis @ X1 @ ( n_pl @ X0 @ n_1 ) ) ) ) )).

thf(satz25a,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( ( d_29_ii @ X1 @ X0 )
           => ( moreis @ X1 @ ( ordsucc @ X0 ) ) ) ) )).

thf(satz25b,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( ( iii @ X1 @ X0 )
           => ( lessis @ ( n_pl @ X1 @ n_1 ) @ X0 ) ) ) )).

thf(satz25c,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( ( iii @ X1 @ X0 )
           => ( lessis @ ( ordsucc @ X1 ) @ X0 ) ) ) )).

thf(satz26,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( ( iii @ X1 @ ( n_pl @ X0 @ n_1 ) )
           => ( lessis @ X1 @ X0 ) ) ) )).

thf(satz26a,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( ( iii @ X1 @ ( ordsucc @ X0 ) )
           => ( lessis @ X1 @ X0 ) ) ) )).

thf(satz26b,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( ( d_29_ii @ ( n_pl @ X1 @ n_1 ) @ X0 )
           => ( moreis @ X1 @ X0 ) ) ) )).

thf(satz26c,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( ( d_29_ii @ ( ordsucc @ X1 ) @ X0 )
           => ( moreis @ X1 @ X0 ) ) ) )).

thf(typ_lbprop,type,(
    lbprop: ( $i > $o ) > $i > $i > $o )).

thf(def_lbprop,definition,
    ( lbprop
    = ( ^ [X0: ( $i > $o ),X1: $i,X2: $i] :
          ( imp @ ( X0 @ X2 ) @ ( lessis @ X1 @ X2 ) ) ) )).

thf(typ_n_lb,type,(
    n_lb: ( $i > $o ) > $i > $o )).

thf(def_n_lb,definition,
    ( n_lb
    = ( ^ [X0: ( $i > $o ),X1: $i] :
          ( n_all @ ( lbprop @ X0 @ X1 ) ) ) )).

thf(typ_min,type,(
    min: ( $i > $o ) > $i > $o )).

thf(def_min,definition,
    ( min
    = ( ^ [X0: ( $i > $o ),X1: $i] :
          ( d_and @ ( n_lb @ X0 @ X1 ) @ ( X0 @ X1 ) ) ) )).

thf(satz27,axiom,(
    ! [X0: ( $i > $o )] :
      ( ( n_some @ X0 )
     => ( n_some @ ( min @ X0 ) ) ) )).

thf(satz27a,axiom,(
    ! [X0: ( $i > $o )] :
      ( ( n_some @ X0 )
     => ( n_one @ ( min @ X0 ) ) ) )).

thf(typ_d_428_prop1,type,(
    d_428_prop1: $i > $i > $o )).

thf(def_d_428_prop1,definition,
    ( d_428_prop1
    = ( ^ [X0: $i,X1: $i] :
          ( n_all
          @ ^ [X2: $i] :
              ( n_is @ ( ap @ X1 @ ( ordsucc @ X2 ) ) @ ( n_pl @ ( ap @ X1 @ X2 ) @ X0 ) ) ) ) )).

thf(typ_d_428_prop2,type,(
    d_428_prop2: $i > $i > $o )).

thf(def_d_428_prop2,definition,
    ( d_428_prop2
    = ( ^ [X0: $i,X1: $i] :
          ( d_and @ ( n_is @ ( ap @ X1 @ n_1 ) @ X0 ) @ ( d_428_prop1 @ X0 @ X1 ) ) ) )).

thf(typ_d_428_prop4,type,(
    d_428_prop4: $i > $o )).

thf(def_d_428_prop4,definition,
    ( d_428_prop4
    = ( ^ [X0: $i] :
          ( l_some
          @ ( d_Pi @ nat
            @ ^ [X1: $i] : nat )
          @ ( d_428_prop2 @ X0 ) ) ) )).

thf(typ_d_428_id,type,(
    d_428_id: $i )).

thf(def_d_428_id,definition,
    ( d_428_id
    = ( d_Sigma @ nat
      @ ^ [X0: $i] : X0 ) )).

thf(typ_d_428_g,type,(
    d_428_g: $i > $i )).

thf(def_d_428_g,definition,
    ( d_428_g
    = ( ^ [X0: $i] :
          ( d_Sigma @ nat
          @ ^ [X1: $i] :
              ( n_pl @ ( ap @ X0 @ X1 ) @ X1 ) ) ) )).

thf(satz28,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( one
        @ ( d_Pi @ nat
          @ ^ [X1: $i] : nat )
        @ ^ [X1: $i] :
            ( d_and @ ( n_is @ ( ap @ X1 @ n_1 ) @ X0 )
            @ ( n_all
              @ ^ [X2: $i] :
                  ( n_is @ ( ap @ X1 @ ( ordsucc @ X2 ) ) @ ( n_pl @ ( ap @ X1 @ X2 ) @ X0 ) ) ) ) ) )).

thf(typ_times,type,(
    times: $i > $i )).

thf(def_times,definition,
    ( times
    = ( ^ [X0: $i] :
          ( ind
          @ ( d_Pi @ nat
            @ ^ [X1: $i] : nat )
          @ ( d_428_prop2 @ X0 ) ) ) )).

thf(typ_n_ts,type,(
    n_ts: $i > $i > $i )).

thf(def_n_ts,definition,
    ( n_ts
    = ( ^ [X0: $i] :
          ( ap @ ( times @ X0 ) ) ) )).

thf(satz28a,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( n_is @ ( n_ts @ X0 @ n_1 ) @ X0 ) )).

thf(satz28b,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( n_is @ ( n_ts @ X0 @ ( ordsucc @ X1 ) ) @ ( n_pl @ ( n_ts @ X0 @ X1 ) @ X0 ) ) ) )).

thf(satz28c,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( n_is @ ( n_ts @ n_1 @ X0 ) @ X0 ) )).

thf(satz28d,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( n_is @ ( n_ts @ ( ordsucc @ X0 ) @ X1 ) @ ( n_pl @ ( n_ts @ X0 @ X1 ) @ X1 ) ) ) )).

thf(satz28e,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( n_is @ X0 @ ( n_ts @ X0 @ n_1 ) ) )).

thf(satz28f,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( n_is @ ( n_pl @ ( n_ts @ X0 @ X1 ) @ X0 ) @ ( n_ts @ X0 @ ( ordsucc @ X1 ) ) ) ) )).

thf(satz28g,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( n_is @ X0 @ ( n_ts @ n_1 @ X0 ) ) )).

thf(satz28h,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( n_is @ ( n_pl @ ( n_ts @ X0 @ X1 ) @ X1 ) @ ( n_ts @ ( ordsucc @ X0 ) @ X1 ) ) ) )).

thf(typ_d_429_prop1,type,(
    d_429_prop1: $i > $i > $o )).

thf(def_d_429_prop1,definition,
    ( d_429_prop1
    = ( ^ [X0: $i,X1: $i] :
          ( n_is @ ( n_ts @ X0 @ X1 ) @ ( n_ts @ X1 @ X0 ) ) ) )).

thf(satz29,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( n_is @ ( n_ts @ X0 @ X1 ) @ ( n_ts @ X1 @ X0 ) ) ) )).

thf(typ_d_430_prop1,type,(
    d_430_prop1: $i > $i > $i > $o )).

thf(def_d_430_prop1,definition,
    ( d_430_prop1
    = ( ^ [X0: $i,X1: $i,X2: $i] :
          ( n_is @ ( n_ts @ X0 @ ( n_pl @ X1 @ X2 ) ) @ ( n_pl @ ( n_ts @ X0 @ X1 ) @ ( n_ts @ X0 @ X2 ) ) ) ) )).

thf(satz30,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( n_is @ ( n_ts @ X0 @ ( n_pl @ X1 @ X2 ) ) @ ( n_pl @ ( n_ts @ X0 @ X1 ) @ ( n_ts @ X0 @ X2 ) ) ) ) ) )).

thf(typ_d_431_prop1,type,(
    d_431_prop1: $i > $i > $i > $o )).

thf(def_d_431_prop1,definition,
    ( d_431_prop1
    = ( ^ [X0: $i,X1: $i,X2: $i] :
          ( n_is @ ( n_ts @ ( n_ts @ X0 @ X1 ) @ X2 ) @ ( n_ts @ X0 @ ( n_ts @ X1 @ X2 ) ) ) ) )).

thf(satz31,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( n_is @ ( n_ts @ ( n_ts @ X0 @ X1 ) @ X2 ) @ ( n_ts @ X0 @ ( n_ts @ X1 @ X2 ) ) ) ) ) )).

thf(satz32a,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( ( d_29_ii @ X0 @ X1 )
               => ( d_29_ii @ ( n_ts @ X0 @ X2 ) @ ( n_ts @ X1 @ X2 ) ) ) ) ) )).

thf(satz32b,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( ( n_is @ X0 @ X1 )
               => ( n_is @ ( n_ts @ X0 @ X2 ) @ ( n_ts @ X1 @ X2 ) ) ) ) ) )).

thf(satz32c,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( ( iii @ X0 @ X1 )
               => ( iii @ ( n_ts @ X0 @ X2 ) @ ( n_ts @ X1 @ X2 ) ) ) ) ) )).

thf(satz32d,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( ( d_29_ii @ X0 @ X1 )
               => ( d_29_ii @ ( n_ts @ X2 @ X0 ) @ ( n_ts @ X2 @ X1 ) ) ) ) ) )).

thf(satz32e,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( ( n_is @ X0 @ X1 )
               => ( n_is @ ( n_ts @ X2 @ X0 ) @ ( n_ts @ X2 @ X1 ) ) ) ) ) )).

thf(satz32f,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( ( iii @ X0 @ X1 )
               => ( iii @ ( n_ts @ X2 @ X0 ) @ ( n_ts @ X2 @ X1 ) ) ) ) ) )).

thf(satz32g,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ nat )
                @ ^ [X3: $i] :
                    ( ( n_is @ X0 @ X1 )
                   => ( ( d_29_ii @ X2 @ X3 )
                     => ( d_29_ii @ ( n_ts @ X0 @ X2 ) @ ( n_ts @ X1 @ X3 ) ) ) ) ) ) ) )).

thf(satz32h,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ nat )
                @ ^ [X3: $i] :
                    ( ( n_is @ X0 @ X1 )
                   => ( ( d_29_ii @ X2 @ X3 )
                     => ( d_29_ii @ ( n_ts @ X2 @ X0 ) @ ( n_ts @ X3 @ X1 ) ) ) ) ) ) ) )).

thf(satz32j,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ nat )
                @ ^ [X3: $i] :
                    ( ( n_is @ X0 @ X1 )
                   => ( ( iii @ X2 @ X3 )
                     => ( iii @ ( n_ts @ X0 @ X2 ) @ ( n_ts @ X1 @ X3 ) ) ) ) ) ) ) )).

thf(satz32k,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ nat )
                @ ^ [X3: $i] :
                    ( ( n_is @ X0 @ X1 )
                   => ( ( iii @ X2 @ X3 )
                     => ( iii @ ( n_ts @ X2 @ X0 ) @ ( n_ts @ X3 @ X1 ) ) ) ) ) ) ) )).

thf(satz32l,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( ( moreis @ X0 @ X1 )
               => ( moreis @ ( n_ts @ X0 @ X2 ) @ ( n_ts @ X1 @ X2 ) ) ) ) ) )).

thf(satz32m,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( ( moreis @ X0 @ X1 )
               => ( moreis @ ( n_ts @ X2 @ X0 ) @ ( n_ts @ X2 @ X1 ) ) ) ) ) )).

thf(satz32n,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( ( lessis @ X0 @ X1 )
               => ( lessis @ ( n_ts @ X0 @ X2 ) @ ( n_ts @ X1 @ X2 ) ) ) ) ) )).

thf(satz32o,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( ( lessis @ X0 @ X1 )
               => ( lessis @ ( n_ts @ X2 @ X0 ) @ ( n_ts @ X2 @ X1 ) ) ) ) ) )).

thf(satz33a,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( ( d_29_ii @ ( n_ts @ X0 @ X2 ) @ ( n_ts @ X1 @ X2 ) )
               => ( d_29_ii @ X0 @ X1 ) ) ) ) )).

thf(satz33b,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( ( n_is @ ( n_ts @ X0 @ X2 ) @ ( n_ts @ X1 @ X2 ) )
               => ( n_is @ X0 @ X1 ) ) ) ) )).

thf(satz33c,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( ( iii @ ( n_ts @ X0 @ X2 ) @ ( n_ts @ X1 @ X2 ) )
               => ( iii @ X0 @ X1 ) ) ) ) )).

thf(satz34,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ nat )
                @ ^ [X3: $i] :
                    ( ( d_29_ii @ X0 @ X1 )
                   => ( ( d_29_ii @ X2 @ X3 )
                     => ( d_29_ii @ ( n_ts @ X0 @ X2 ) @ ( n_ts @ X1 @ X3 ) ) ) ) ) ) ) )).

thf(satz34a,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ nat )
                @ ^ [X3: $i] :
                    ( ( iii @ X0 @ X1 )
                   => ( ( iii @ X2 @ X3 )
                     => ( iii @ ( n_ts @ X0 @ X2 ) @ ( n_ts @ X1 @ X3 ) ) ) ) ) ) ) )).

thf(satz35a,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ nat )
                @ ^ [X3: $i] :
                    ( ( moreis @ X0 @ X1 )
                   => ( ( d_29_ii @ X2 @ X3 )
                     => ( d_29_ii @ ( n_ts @ X0 @ X2 ) @ ( n_ts @ X1 @ X3 ) ) ) ) ) ) ) )).

thf(satz35b,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ nat )
                @ ^ [X3: $i] :
                    ( ( d_29_ii @ X0 @ X1 )
                   => ( ( moreis @ X2 @ X3 )
                     => ( d_29_ii @ ( n_ts @ X0 @ X2 ) @ ( n_ts @ X1 @ X3 ) ) ) ) ) ) ) )).

thf(satz35c,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ nat )
                @ ^ [X3: $i] :
                    ( ( lessis @ X0 @ X1 )
                   => ( ( iii @ X2 @ X3 )
                     => ( iii @ ( n_ts @ X0 @ X2 ) @ ( n_ts @ X1 @ X3 ) ) ) ) ) ) ) )).

thf(satz35d,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ nat )
                @ ^ [X3: $i] :
                    ( ( iii @ X0 @ X1 )
                   => ( ( lessis @ X2 @ X3 )
                     => ( iii @ ( n_ts @ X0 @ X2 ) @ ( n_ts @ X1 @ X3 ) ) ) ) ) ) ) )).

thf(satz36,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ nat )
                @ ^ [X3: $i] :
                    ( ( moreis @ X0 @ X1 )
                   => ( ( moreis @ X2 @ X3 )
                     => ( moreis @ ( n_ts @ X0 @ X2 ) @ ( n_ts @ X1 @ X3 ) ) ) ) ) ) ) )).

thf(satz36a,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ nat )
                @ ^ [X3: $i] :
                    ( ( lessis @ X0 @ X1 )
                   => ( ( lessis @ X2 @ X3 )
                     => ( lessis @ ( n_ts @ X0 @ X2 ) @ ( n_ts @ X1 @ X3 ) ) ) ) ) ) ) )).

thf(typ_n_mn,type,(
    n_mn: $i > $i > $i )).

thf(def_n_mn,definition,
    ( n_mn
    = ( ^ [X0: $i,X1: $i] :
          ( ind @ nat @ ( diffprop @ X0 @ X1 ) ) ) )).

thf(typ_d_1to,type,(
    d_1to: $i > $i )).

thf(def_d_1to,definition,
    ( d_1to
    = ( ^ [X0: $i] :
          ( d_Sep @ nat
          @ ^ [X1: $i] :
              ( lessis @ X1 @ X0 ) ) ) )).

thf(typ_outn,type,(
    outn: $i > $i > $i )).

thf(def_outn,definition,
    ( outn
    = ( ^ [X0: $i] :
          ( out @ nat
          @ ^ [X1: $i] :
              ( lessis @ X1 @ X0 ) ) ) )).

thf(typ_inn,type,(
    inn: $i > $i > $i )).

thf(def_inn,definition,
    ( inn
    = ( ^ [X0: $i] :
          ( e_in @ nat
          @ ^ [X1: $i] :
              ( lessis @ X1 @ X0 ) ) ) )).

thf(typ_n_1o,type,(
    n_1o: $i )).

thf(def_n_1o,definition,
    ( n_1o
    = ( outn @ n_1 @ n_1 ) )).

thf(typ_singlet_u0,type,(
    singlet_u0: $i > $i )).

thf(def_singlet_u0,definition,
    ( singlet_u0
    = ( inn @ n_1 ) )).

thf(typ_n_2,type,(
    n_2: $i )).

thf(def_n_2,definition,
    ( n_2
    = ( n_pl @ n_1 @ n_1 ) )).

thf(typ_n_1t,type,(
    n_1t: $i )).

thf(def_n_1t,definition,
    ( n_1t
    = ( outn @ n_2 @ n_1 ) )).

thf(typ_n_2t,type,(
    n_2t: $i )).

thf(def_n_2t,definition,
    ( n_2t
    = ( outn @ n_2 @ n_2 ) )).

thf(typ_pair_u0,type,(
    pair_u0: $i > $i )).

thf(def_pair_u0,definition,
    ( pair_u0
    = ( inn @ n_2 ) )).

thf(typ_pair1type,type,(
    pair1type: $i > $i )).

thf(def_pair1type,definition,
    ( pair1type
    = ( ^ [X0: $i] :
          ( d_Pi @ ( d_1to @ n_2 )
          @ ^ [X1: $i] : X0 ) ) )).

thf(typ_pair1,type,(
    pair1: $i > $i > $i > $i )).

thf(def_pair1,definition,
    ( pair1
    = ( ^ [X0: $i,X1: $i,X2: $i] :
          ( d_Sigma @ ( d_1to @ n_2 )
          @ ^ [X3: $i] :
              ( ite @ ( e_is @ ( d_1to @ n_2 ) @ X3 @ n_1t ) @ X0 @ X1 @ X2 ) ) ) )).

thf(typ_first1,type,(
    first1: $i > $i > $i )).

thf(def_first1,definition,
    ( first1
    = ( ^ [X0: $i,X1: $i] :
          ( ap @ X1 @ n_1t ) ) )).

thf(typ_second1,type,(
    second1: $i > $i > $i )).

thf(def_second1,definition,
    ( second1
    = ( ^ [X0: $i,X1: $i] :
          ( ap @ X1 @ n_2t ) ) )).

thf(typ_pair_q0,type,(
    pair_q0: $i > $i > $i )).

thf(def_pair_q0,definition,
    ( pair_q0
    = ( ^ [X0: $i,X1: $i] :
          ( pair1 @ X0 @ ( first1 @ X0 @ X1 ) @ ( second1 @ X0 @ X1 ) ) ) )).

thf(typ_d_1out,type,(
    d_1out: $i > $i )).

thf(def_d_1out,definition,
    ( d_1out
    = ( ^ [X0: $i] :
          ( outn @ X0 @ n_1 ) ) )).

thf(typ_xout,type,(
    xout: $i > $i )).

thf(def_xout,definition,
    ( xout
    = ( ^ [X0: $i] :
          ( outn @ X0 @ X0 ) ) )).

thf(typ_left1to,type,(
    left1to: $i > $i > $i > $i )).

thf(def_left1to,definition,
    ( left1to
    = ( ^ [X0: $i,X1: $i,X2: $i] :
          ( outn @ X0 @ ( inn @ X1 @ X2 ) ) ) )).

thf(typ_right1to,type,(
    right1to: $i > $i > $i > $i )).

thf(def_right1to,definition,
    ( right1to
    = ( ^ [X0: $i,X1: $i,X2: $i] :
          ( outn @ ( n_pl @ X0 @ X1 ) @ ( n_pl @ X0 @ ( inn @ X1 @ X2 ) ) ) ) )).

thf(typ_left,type,(
    left: $i > $i > $i > $i > $i )).

thf(def_left,definition,
    ( left
    = ( ^ [X0: $i,X1: $i,X2: $i,X3: $i] :
          ( d_Sigma @ ( d_1to @ X2 )
          @ ^ [X4: $i] :
              ( ap @ X3 @ ( left1to @ X1 @ X2 @ X4 ) ) ) ) )).

thf(typ_right,type,(
    right: $i > $i > $i > $i > $i )).

thf(def_right,definition,
    ( right
    = ( ^ [X0: $i,X1: $i,X2: $i,X3: $i] :
          ( d_Sigma @ ( d_1to @ X2 )
          @ ^ [X4: $i] :
              ( ap @ X3 @ ( right1to @ X1 @ X2 @ X4 ) ) ) ) )).

thf(typ_left_f1,type,(
    left_f1: $i > $i > $i > $i > $i )).

thf(def_left_f1,definition,
    ( left_f1
    = ( ^ [X0: $i,X1: $i,X2: $i] :
          ( left @ X0 @ X2 @ X1 ) ) )).

thf(typ_left_f2,type,(
    left_f2: $i > $i > $i > $i > $i )).

thf(def_left_f2,definition,
    ( left_f2
    = ( ^ [X0: $i,X1: $i,X2: $i,X3: $i] :
          ( left @ X0 @ X1 @ X2 @ ( left_f1 @ X0 @ X1 @ X2 @ X3 ) ) ) )).

thf(typ_frac,type,(
    frac: $i )).

thf(def_frac,definition,
    ( frac
    = ( pair1type @ nat ) )).

thf(typ_n_fr,type,(
    n_fr: $i > $i > $i )).

thf(def_n_fr,definition,
    ( n_fr
    = ( pair1 @ nat ) )).

thf(typ_num,type,(
    num: $i > $i )).

thf(def_num,definition,
    ( num
    = ( first1 @ nat ) )).

thf(typ_den,type,(
    den: $i > $i )).

thf(def_den,definition,
    ( den
    = ( second1 @ nat ) )).

thf(typ_n_eq,type,(
    n_eq: $i > $i > $o )).

thf(def_n_eq,definition,
    ( n_eq
    = ( ^ [X0: $i,X1: $i] :
          ( n_is @ ( n_ts @ ( num @ X0 ) @ ( den @ X1 ) ) @ ( n_ts @ ( num @ X1 ) @ ( den @ X0 ) ) ) ) )).

thf(satz37,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( n_eq @ X0 @ X0 ) )).

thf(satz38,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ frac )
        @ ^ [X1: $i] :
            ( ( n_eq @ X0 @ X1 )
           => ( n_eq @ X1 @ X0 ) ) ) )).

thf(satz39,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ frac )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ frac )
            @ ^ [X2: $i] :
                ( ( n_eq @ X0 @ X1 )
               => ( ( n_eq @ X1 @ X2 )
                 => ( n_eq @ X0 @ X2 ) ) ) ) ) )).

thf(satz40,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( n_eq @ X0 @ ( n_fr @ ( n_ts @ ( num @ X0 ) @ X1 ) @ ( n_ts @ ( den @ X0 ) @ X1 ) ) ) ) )).

thf(satz40a,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( n_eq @ ( n_fr @ ( n_ts @ ( num @ X0 ) @ X1 ) @ ( n_ts @ ( den @ X0 ) @ X1 ) ) @ X0 ) ) )).

thf(satz40b,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( n_eq @ ( n_fr @ X0 @ X1 ) @ ( n_fr @ ( n_ts @ X0 @ X2 ) @ ( n_ts @ X1 @ X2 ) ) ) ) ) )).

thf(satz40c,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( n_eq @ ( n_fr @ ( n_ts @ X0 @ X2 ) @ ( n_ts @ X1 @ X2 ) ) @ ( n_fr @ X0 @ X1 ) ) ) ) )).

thf(typ_moref,type,(
    moref: $i > $i > $o )).

thf(def_moref,definition,
    ( moref
    = ( ^ [X0: $i,X1: $i] :
          ( d_29_ii @ ( n_ts @ ( num @ X0 ) @ ( den @ X1 ) ) @ ( n_ts @ ( num @ X1 ) @ ( den @ X0 ) ) ) ) )).

thf(typ_lessf,type,(
    lessf: $i > $i > $o )).

thf(def_lessf,definition,
    ( lessf
    = ( ^ [X0: $i,X1: $i] :
          ( iii @ ( n_ts @ ( num @ X0 ) @ ( den @ X1 ) ) @ ( n_ts @ ( num @ X1 ) @ ( den @ X0 ) ) ) ) )).

thf(satz41,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ frac )
        @ ^ [X1: $i] :
            ( orec3 @ ( n_eq @ X0 @ X1 ) @ ( moref @ X0 @ X1 ) @ ( lessf @ X0 @ X1 ) ) ) )).

thf(satz41a,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ frac )
        @ ^ [X1: $i] :
            ( or3 @ ( n_eq @ X0 @ X1 ) @ ( moref @ X0 @ X1 ) @ ( lessf @ X0 @ X1 ) ) ) )).

thf(satz41b,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ frac )
        @ ^ [X1: $i] :
            ( ec3 @ ( n_eq @ X0 @ X1 ) @ ( moref @ X0 @ X1 ) @ ( lessf @ X0 @ X1 ) ) ) )).

thf(satz42,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ frac )
        @ ^ [X1: $i] :
            ( ( moref @ X0 @ X1 )
           => ( lessf @ X1 @ X0 ) ) ) )).

thf(satz43,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ frac )
        @ ^ [X1: $i] :
            ( ( lessf @ X0 @ X1 )
           => ( moref @ X1 @ X0 ) ) ) )).

thf(satz44,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ frac )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ frac )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ frac )
                @ ^ [X3: $i] :
                    ( ( moref @ X0 @ X1 )
                   => ( ( n_eq @ X0 @ X2 )
                     => ( ( n_eq @ X1 @ X3 )
                       => ( moref @ X2 @ X3 ) ) ) ) ) ) ) )).

thf(satz45,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ frac )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ frac )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ frac )
                @ ^ [X3: $i] :
                    ( ( lessf @ X0 @ X1 )
                   => ( ( n_eq @ X0 @ X2 )
                     => ( ( n_eq @ X1 @ X3 )
                       => ( lessf @ X2 @ X3 ) ) ) ) ) ) ) )).

thf(typ_moreq,type,(
    moreq: $i > $i > $o )).

thf(def_moreq,definition,
    ( moreq
    = ( ^ [X0: $i,X1: $i] :
          ( l_or @ ( moref @ X0 @ X1 ) @ ( n_eq @ X0 @ X1 ) ) ) )).

thf(typ_lesseq,type,(
    lesseq: $i > $i > $o )).

thf(def_lesseq,definition,
    ( lesseq
    = ( ^ [X0: $i,X1: $i] :
          ( l_or @ ( lessf @ X0 @ X1 ) @ ( n_eq @ X0 @ X1 ) ) ) )).

thf(satz41c,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ frac )
        @ ^ [X1: $i] :
            ( ( moreq @ X0 @ X1 )
           => ( d_not @ ( lessf @ X0 @ X1 ) ) ) ) )).

thf(satz41d,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ frac )
        @ ^ [X1: $i] :
            ( ( lesseq @ X0 @ X1 )
           => ( d_not @ ( moref @ X0 @ X1 ) ) ) ) )).

thf(satz41e,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ frac )
        @ ^ [X1: $i] :
            ( ( d_not @ ( moref @ X0 @ X1 ) )
           => ( lesseq @ X0 @ X1 ) ) ) )).

thf(satz41f,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ frac )
        @ ^ [X1: $i] :
            ( ( d_not @ ( lessf @ X0 @ X1 ) )
           => ( moreq @ X0 @ X1 ) ) ) )).

thf(satz41g,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ frac )
        @ ^ [X1: $i] :
            ( ( moref @ X0 @ X1 )
           => ( d_not @ ( lesseq @ X0 @ X1 ) ) ) ) )).

thf(satz41h,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ frac )
        @ ^ [X1: $i] :
            ( ( lessf @ X0 @ X1 )
           => ( d_not @ ( moreq @ X0 @ X1 ) ) ) ) )).

thf(satz41j,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ frac )
        @ ^ [X1: $i] :
            ( ( d_not @ ( moreq @ X0 @ X1 ) )
           => ( lessf @ X0 @ X1 ) ) ) )).

thf(satz41k,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ frac )
        @ ^ [X1: $i] :
            ( ( d_not @ ( lesseq @ X0 @ X1 ) )
           => ( moref @ X0 @ X1 ) ) ) )).

thf(satz46,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ frac )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ frac )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ frac )
                @ ^ [X3: $i] :
                    ( ( moreq @ X0 @ X1 )
                   => ( ( n_eq @ X0 @ X2 )
                     => ( ( n_eq @ X1 @ X3 )
                       => ( moreq @ X2 @ X3 ) ) ) ) ) ) ) )).

thf(satz47,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ frac )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ frac )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ frac )
                @ ^ [X3: $i] :
                    ( ( lesseq @ X0 @ X1 )
                   => ( ( n_eq @ X0 @ X2 )
                     => ( ( n_eq @ X1 @ X3 )
                       => ( lesseq @ X2 @ X3 ) ) ) ) ) ) ) )).

thf(satz48,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ frac )
        @ ^ [X1: $i] :
            ( ( moreq @ X0 @ X1 )
           => ( lesseq @ X1 @ X0 ) ) ) )).

thf(satz49,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ frac )
        @ ^ [X1: $i] :
            ( ( lesseq @ X0 @ X1 )
           => ( moreq @ X1 @ X0 ) ) ) )).

thf(satz50,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ frac )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ frac )
            @ ^ [X2: $i] :
                ( ( lessf @ X0 @ X1 )
               => ( ( lessf @ X1 @ X2 )
                 => ( lessf @ X0 @ X2 ) ) ) ) ) )).

thf(satz51a,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ frac )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ frac )
            @ ^ [X2: $i] :
                ( ( lesseq @ X0 @ X1 )
               => ( ( lessf @ X1 @ X2 )
                 => ( lessf @ X0 @ X2 ) ) ) ) ) )).

thf(satz51b,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ frac )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ frac )
            @ ^ [X2: $i] :
                ( ( lessf @ X0 @ X1 )
               => ( ( lesseq @ X1 @ X2 )
                 => ( lessf @ X0 @ X2 ) ) ) ) ) )).

thf(satz51c,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ frac )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ frac )
            @ ^ [X2: $i] :
                ( ( moreq @ X0 @ X1 )
               => ( ( moref @ X1 @ X2 )
                 => ( moref @ X0 @ X2 ) ) ) ) ) )).

thf(satz51d,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ frac )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ frac )
            @ ^ [X2: $i] :
                ( ( moref @ X0 @ X1 )
               => ( ( moreq @ X1 @ X2 )
                 => ( moref @ X0 @ X2 ) ) ) ) ) )).

thf(satz52,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ frac )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ frac )
            @ ^ [X2: $i] :
                ( ( lesseq @ X0 @ X1 )
               => ( ( lesseq @ X1 @ X2 )
                 => ( lesseq @ X0 @ X2 ) ) ) ) ) )).

thf(satz53,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( l_some @ frac
        @ ^ [X1: $i] :
            ( moref @ X1 @ X0 ) ) )).

thf(satz54,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( l_some @ frac
        @ ^ [X1: $i] :
            ( lessf @ X1 @ X0 ) ) )).

thf(satz55,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ frac )
        @ ^ [X1: $i] :
            ( ( lessf @ X0 @ X1 )
           => ( l_some @ frac
              @ ^ [X2: $i] :
                  ( d_and @ ( lessf @ X0 @ X2 ) @ ( lessf @ X2 @ X1 ) ) ) ) ) )).

thf(typ_n_pf,type,(
    n_pf: $i > $i > $i )).

thf(def_n_pf,definition,
    ( n_pf
    = ( ^ [X0: $i,X1: $i] :
          ( n_fr @ ( n_pl @ ( n_ts @ ( num @ X0 ) @ ( den @ X1 ) ) @ ( n_ts @ ( num @ X1 ) @ ( den @ X0 ) ) ) @ ( n_ts @ ( den @ X0 ) @ ( den @ X1 ) ) ) ) )).

thf(satz56,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ frac )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ frac )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ frac )
                @ ^ [X3: $i] :
                    ( ( n_eq @ X0 @ X1 )
                   => ( ( n_eq @ X2 @ X3 )
                     => ( n_eq @ ( n_pf @ X0 @ X2 ) @ ( n_pf @ X1 @ X3 ) ) ) ) ) ) ) )).

thf(satz57,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( n_eq @ ( n_pf @ ( n_fr @ X0 @ X2 ) @ ( n_fr @ X1 @ X2 ) ) @ ( n_fr @ ( n_pl @ X0 @ X1 ) @ X2 ) ) ) ) )).

thf(satz57a,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ nat )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ nat )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ nat )
            @ ^ [X2: $i] :
                ( n_eq @ ( n_fr @ ( n_pl @ X0 @ X1 ) @ X2 ) @ ( n_pf @ ( n_fr @ X0 @ X2 ) @ ( n_fr @ X1 @ X2 ) ) ) ) ) )).

thf(satz58,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ frac )
        @ ^ [X1: $i] :
            ( n_eq @ ( n_pf @ X0 @ X1 ) @ ( n_pf @ X1 @ X0 ) ) ) )).

thf(satz59,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ frac )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ frac )
            @ ^ [X2: $i] :
                ( n_eq @ ( n_pf @ ( n_pf @ X0 @ X1 ) @ X2 ) @ ( n_pf @ X0 @ ( n_pf @ X1 @ X2 ) ) ) ) ) )).

thf(satz60,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ frac )
        @ ^ [X1: $i] :
            ( moref @ ( n_pf @ X0 @ X1 ) @ X0 ) ) )).

thf(satz60a,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ frac )
        @ ^ [X1: $i] :
            ( lessf @ X0 @ ( n_pf @ X0 @ X1 ) ) ) )).

thf(satz61,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ frac )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ frac )
            @ ^ [X2: $i] :
                ( ( moref @ X0 @ X1 )
               => ( moref @ ( n_pf @ X0 @ X2 ) @ ( n_pf @ X1 @ X2 ) ) ) ) ) )).

thf(satz62b,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ frac )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ frac )
            @ ^ [X2: $i] :
                ( ( n_eq @ X0 @ X1 )
               => ( n_eq @ ( n_pf @ X0 @ X2 ) @ ( n_pf @ X1 @ X2 ) ) ) ) ) )).

thf(satz62c,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ frac )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ frac )
            @ ^ [X2: $i] :
                ( ( lessf @ X0 @ X1 )
               => ( lessf @ ( n_pf @ X0 @ X2 ) @ ( n_pf @ X1 @ X2 ) ) ) ) ) )).

thf(satz62d,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ frac )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ frac )
            @ ^ [X2: $i] :
                ( ( moref @ X0 @ X1 )
               => ( moref @ ( n_pf @ X2 @ X0 ) @ ( n_pf @ X2 @ X1 ) ) ) ) ) )).

thf(satz62e,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ frac )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ frac )
            @ ^ [X2: $i] :
                ( ( n_eq @ X0 @ X1 )
               => ( n_eq @ ( n_pf @ X2 @ X0 ) @ ( n_pf @ X2 @ X1 ) ) ) ) ) )).

thf(satz62f,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ frac )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ frac )
            @ ^ [X2: $i] :
                ( ( lessf @ X0 @ X1 )
               => ( lessf @ ( n_pf @ X2 @ X0 ) @ ( n_pf @ X2 @ X1 ) ) ) ) ) )).

thf(satz62g,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ frac )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ frac )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ frac )
                @ ^ [X3: $i] :
                    ( ( n_eq @ X0 @ X1 )
                   => ( ( moref @ X2 @ X3 )
                     => ( moref @ ( n_pf @ X0 @ X2 ) @ ( n_pf @ X1 @ X3 ) ) ) ) ) ) ) )).

thf(satz62h,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ frac )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ frac )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ frac )
                @ ^ [X3: $i] :
                    ( ( n_eq @ X0 @ X1 )
                   => ( ( moref @ X2 @ X3 )
                     => ( moref @ ( n_pf @ X2 @ X0 ) @ ( n_pf @ X3 @ X1 ) ) ) ) ) ) ) )).

thf(satz62j,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ frac )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ frac )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ frac )
                @ ^ [X3: $i] :
                    ( ( n_eq @ X0 @ X1 )
                   => ( ( lessf @ X2 @ X3 )
                     => ( lessf @ ( n_pf @ X0 @ X2 ) @ ( n_pf @ X1 @ X3 ) ) ) ) ) ) ) )).

thf(satz62k,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ frac )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ frac )
            @ ^ [X2: $i] :
                ( all_of
                @ ^ [X3: $i] :
                    ( in @ X3 @ frac )
                @ ^ [X3: $i] :
                    ( ( n_eq @ X0 @ X1 )
                   => ( ( lessf @ X2 @ X3 )
                     => ( lessf @ ( n_pf @ X2 @ X0 ) @ ( n_pf @ X3 @ X1 ) ) ) ) ) ) ) )).

thf(satz63a,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ frac )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ frac )
            @ ^ [X2: $i] :
                ( ( moref @ ( n_pf @ X0 @ X2 ) @ ( n_pf @ X1 @ X2 ) )
               => ( moref @ X0 @ X1 ) ) ) ) )).

thf(satz63b,axiom,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ frac )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ frac )
            @ ^ [X2: $i] :
                ( ( n_eq @ ( n_pf @ X0 @ X2 ) @ ( n_pf @ X1 @ X2 ) )
               => ( n_eq @ X0 @ X1 ) ) ) ) )).

thf(satz63c,conjecture,
    ( all_of
    @ ^ [X0: $i] :
        ( in @ X0 @ frac )
    @ ^ [X0: $i] :
        ( all_of
        @ ^ [X1: $i] :
            ( in @ X1 @ frac )
        @ ^ [X1: $i] :
            ( all_of
            @ ^ [X2: $i] :
                ( in @ X2 @ frac )
            @ ^ [X2: $i] :
                ( ( lessf @ ( n_pf @ X0 @ X2 ) @ ( n_pf @ X1 @ X2 ) )
               => ( lessf @ X0 @ X1 ) ) ) ) )).

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