SET007 Axioms: SET007+125.ax


%------------------------------------------------------------------------------
% File     : SET007+125 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Cartesian Product of Functions
% Version  : [Urb08] axioms.
% English  :

% Refs     : [Mat90] Matuszewski (1990), Formalized Mathematics
%          : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
%          : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source   : [Urb08]
% Names    : funct_6 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  114 (  17 unit)
%            Number of atoms       :  574 ( 155 equality)
%            Maximal formula depth :   20 (   7 average)
%            Number of connectives :  493 (  33 ~  ;  11  |; 243  &)
%                                         (  17 <=>; 189 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   15 (   1 propositional; 0-3 arity)
%            Number of functors    :   52 (   6 constant; 0-3 arity)
%            Number of variables   :  303 (   0 singleton; 292 !;  11 ?)
%            Maximal term depth    :    6 (   2 average)
% SPC      : 

% Comments : The individual reference can be found in [Mat90] by looking for
%            the name provided by [Urb08].
%          : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
%          : These set theory axioms are used in encodings of problems in
%            various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(t1_funct_6,axiom,(
    ! [A,B] :
      ( r2_hidden(A,k4_card_3(k9_finseq_1(B)))
    <=> ? [C] :
          ( r2_hidden(C,B)
          & A = k9_finseq_1(C) ) ) )).

fof(t2_funct_6,axiom,(
    ! [A,B,C] :
      ( r2_hidden(A,k4_card_3(k10_finseq_1(B,C)))
    <=> ? [D,E] :
          ( r2_hidden(D,B)
          & r2_hidden(E,C)
          & A = k10_finseq_1(D,E) ) ) )).

fof(t3_funct_6,axiom,(
    ! [A,B,C,D] :
      ( r2_hidden(A,k4_card_3(k11_finseq_1(B,C,D)))
    <=> ? [E,F,G] :
          ( r2_hidden(E,B)
          & r2_hidden(F,C)
          & r2_hidden(G,D)
          & A = k11_finseq_1(E,F,G) ) ) )).

fof(t4_funct_6,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => k4_card_3(k9_finseq_1(A)) = k4_finseq_2(np__1,A) ) )).

fof(t6_funct_6,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => k4_card_3(k10_finseq_1(A,A)) = k4_finseq_2(np__2,A) ) )).

fof(t8_funct_6,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => k4_card_3(k11_finseq_1(A,A,A)) = k4_finseq_2(np__3,A) ) )).

fof(t9_funct_6,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => k4_card_3(k2_finseq_2(A,B)) = k4_finseq_2(A,B) ) ) )).

fof(t10_funct_6,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => r1_tarski(k4_card_3(A),k1_funct_2(k1_relat_1(A),k3_card_3(A))) ) )).

fof(t11_funct_6,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ~ ( r2_hidden(A,k1_relat_1(k2_funct_4(B)))
          & ! [C,D] : A != k4_tarski(C,D) ) ) )).

fof(t12_funct_6,axiom,(
    ! [A,B,C] : k2_funct_4(k2_funcop_1(k2_zfmisc_1(A,B),C)) = k2_funcop_1(k2_zfmisc_1(B,A),C) )).

fof(t13_funct_6,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( k3_funct_5(A) = k5_funct_5(k2_funct_4(A))
        & k4_funct_5(A) = k2_funct_4(k6_funct_5(A)) ) ) )).

fof(t14_funct_6,axiom,(
    ! [A,B,C] :
      ( k2_zfmisc_1(A,B) != k1_xboole_0
     => ( k3_funct_5(k2_funcop_1(k2_zfmisc_1(A,B),C)) = k2_funcop_1(A,k2_funcop_1(B,C))
        & k5_funct_5(k2_funcop_1(k2_zfmisc_1(A,B),C)) = k2_funcop_1(B,k2_funcop_1(A,C)) ) ) )).

fof(t15_funct_6,axiom,(
    ! [A,B,C] :
      ( k4_funct_5(k2_funcop_1(A,k2_funcop_1(B,C))) = k2_funcop_1(k2_zfmisc_1(A,B),C)
      & k6_funct_5(k2_funcop_1(A,k2_funcop_1(B,C))) = k2_funcop_1(k2_zfmisc_1(B,A),C) ) )).

fof(t16_funct_6,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ! [C] :
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => ( ( r2_hidden(A,k1_relat_1(B))
              & C = k1_funct_1(B,A) )
           => ( r1_tarski(k2_relat_1(C),k2_relat_1(k4_funct_5(B)))
              & r1_tarski(k2_relat_1(C),k2_relat_1(k6_funct_5(B))) ) ) ) ) )).

fof(t17_funct_6,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( k1_relat_1(k4_funct_5(k2_funcop_1(A,B))) = k2_zfmisc_1(A,k1_relat_1(B))
        & r1_tarski(k2_relat_1(k4_funct_5(k2_funcop_1(A,B))),k2_relat_1(B))
        & k1_relat_1(k6_funct_5(k2_funcop_1(A,B))) = k2_zfmisc_1(k1_relat_1(B),A)
        & r1_tarski(k2_relat_1(k6_funct_5(k2_funcop_1(A,B))),k2_relat_1(B)) ) ) )).

fof(t18_funct_6,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( A != k1_xboole_0
       => ( k2_relat_1(k4_funct_5(k2_funcop_1(A,B))) = k2_relat_1(B)
          & k2_relat_1(k6_funct_5(k2_funcop_1(A,B))) = k2_relat_1(B) ) ) ) )).

fof(t19_funct_6,axiom,(
    ! [A,B,C,D] :
      ( ( v1_relat_1(D)
        & v1_funct_1(D) )
     => ( r2_hidden(D,k1_funct_2(k2_zfmisc_1(A,B),C))
       => ( k2_zfmisc_1(A,B) = k1_xboole_0
          | ( r2_hidden(k3_funct_5(D),k1_funct_2(A,k1_funct_2(B,C)))
            & r2_hidden(k5_funct_5(D),k1_funct_2(B,k1_funct_2(A,C))) ) ) ) ) )).

fof(t20_funct_6,axiom,(
    ! [A,B,C,D] :
      ( ( v1_relat_1(D)
        & v1_funct_1(D) )
     => ( r2_hidden(D,k1_funct_2(A,k1_funct_2(B,C)))
       => ( r2_hidden(k4_funct_5(D),k1_funct_2(k2_zfmisc_1(A,B),C))
          & r2_hidden(k6_funct_5(D),k1_funct_2(k2_zfmisc_1(B,A),C)) ) ) ) )).

fof(t21_funct_6,axiom,(
    ! [A,B,C,D,E,F] :
      ( ( v1_relat_1(F)
        & v1_funct_1(F) )
     => ( r1_tarski(k1_relat_1(F),k2_zfmisc_1(D,E))
       => ( ( ~ r2_hidden(k3_funct_5(F),k1_funct_2(A,k1_funct_2(B,C)))
            & ~ r2_hidden(k5_funct_5(F),k1_funct_2(B,k1_funct_2(A,C))) )
          | r2_hidden(F,k1_funct_2(k2_zfmisc_1(A,B),C)) ) ) ) )).

fof(t22_funct_6,axiom,(
    ! [A,B,C,D,E,F] :
      ( ( v1_relat_1(F)
        & v1_funct_1(F) )
     => ( ( r1_tarski(k2_relat_1(F),k4_partfun1(D,E))
          & k1_relat_1(F) = A )
       => ( ( ~ r2_hidden(k4_funct_5(F),k1_funct_2(k2_zfmisc_1(A,B),C))
            & ~ r2_hidden(k6_funct_5(F),k1_funct_2(k2_zfmisc_1(B,A),C)) )
          | r2_hidden(F,k1_funct_2(A,k1_funct_2(B,C))) ) ) ) )).

fof(t23_funct_6,axiom,(
    ! [A,B,C,D] :
      ( ( v1_relat_1(D)
        & v1_funct_1(D) )
     => ( r2_hidden(D,k4_partfun1(k2_zfmisc_1(A,B),C))
       => ( r2_hidden(k3_funct_5(D),k4_partfun1(A,k4_partfun1(B,C)))
          & r2_hidden(k5_funct_5(D),k4_partfun1(B,k4_partfun1(A,C))) ) ) ) )).

fof(t24_funct_6,axiom,(
    ! [A,B,C,D] :
      ( ( v1_relat_1(D)
        & v1_funct_1(D) )
     => ( r2_hidden(D,k4_partfun1(A,k4_partfun1(B,C)))
       => ( r2_hidden(k4_funct_5(D),k4_partfun1(k2_zfmisc_1(A,B),C))
          & r2_hidden(k6_funct_5(D),k4_partfun1(k2_zfmisc_1(B,A),C)) ) ) ) )).

fof(t25_funct_6,axiom,(
    ! [A,B,C,D,E,F] :
      ( ( v1_relat_1(F)
        & v1_funct_1(F) )
     => ( r1_tarski(k1_relat_1(F),k2_zfmisc_1(D,E))
       => ( ( ~ r2_hidden(k3_funct_5(F),k4_partfun1(A,k4_partfun1(B,C)))
            & ~ r2_hidden(k5_funct_5(F),k4_partfun1(B,k4_partfun1(A,C))) )
          | r2_hidden(F,k4_partfun1(k2_zfmisc_1(A,B),C)) ) ) ) )).

fof(t26_funct_6,axiom,(
    ! [A,B,C,D,E,F] :
      ( ( v1_relat_1(F)
        & v1_funct_1(F) )
     => ( ( r1_tarski(k2_relat_1(F),k4_partfun1(D,E))
          & r1_tarski(k1_relat_1(F),A) )
       => ( ( ~ r2_hidden(k4_funct_5(F),k4_partfun1(k2_zfmisc_1(A,B),C))
            & ~ r2_hidden(k6_funct_5(F),k4_partfun1(k2_zfmisc_1(B,A),C)) )
          | r2_hidden(F,k4_partfun1(A,k4_partfun1(B,C))) ) ) ) )).

fof(d1_funct_6,axiom,(
    ! [A,B] :
      ( B = k1_funct_6(A)
    <=> ! [C] :
          ( r2_hidden(C,B)
        <=> ( r2_hidden(C,A)
            & v1_relat_1(C)
            & v1_funct_1(C) ) ) ) )).

fof(t27_funct_6,axiom,(
    ! [A] : r1_tarski(k1_funct_6(A),A) )).

fof(t28_funct_6,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( r2_hidden(A,k10_relat_1(B,k1_funct_6(k2_relat_1(B))))
      <=> ( r2_hidden(A,k1_relat_1(B))
          & v1_relat_1(k1_funct_1(B,A))
          & v1_funct_1(k1_funct_1(B,A)) ) ) ) )).

fof(t29_funct_6,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ! [C] :
              ( ( v1_relat_1(C)
                & v1_funct_1(C) )
             => ( k1_funct_6(k1_xboole_0) = k1_xboole_0
                & k1_funct_6(k1_tarski(A)) = k1_tarski(A)
                & k1_funct_6(k2_tarski(A,B)) = k2_tarski(A,B)
                & k1_funct_6(k1_enumset1(A,B,C)) = k1_enumset1(A,B,C) ) ) ) ) )).

fof(t30_funct_6,axiom,(
    ! [A,B] :
      ( r1_tarski(A,k1_funct_6(B))
     => k1_funct_6(A) = A ) )).

fof(d2_funct_6,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( B = k2_funct_6(A)
          <=> ( k1_relat_1(B) = k10_relat_1(A,k1_funct_6(k2_relat_1(A)))
              & ! [C] :
                  ( r2_hidden(C,k10_relat_1(A,k1_funct_6(k2_relat_1(A))))
                 => k1_funct_1(B,C) = k1_funct_5(k1_funct_1(A,C)) ) ) ) ) ) )).

fof(d3_funct_6,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( B = k3_funct_6(A)
          <=> ( k1_relat_1(B) = k10_relat_1(A,k1_funct_6(k2_relat_1(A)))
              & ! [C] :
                  ( r2_hidden(C,k10_relat_1(A,k1_funct_6(k2_relat_1(A))))
                 => k1_funct_1(B,C) = k2_funct_5(k1_funct_1(A,C)) ) ) ) ) ) )).

fof(d4_funct_6,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => k4_funct_6(A) = k1_setfam_1(k2_relat_1(A)) ) )).

fof(t31_funct_6,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ! [C] :
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => ( ( r2_hidden(A,k1_relat_1(B))
              & C = k1_funct_1(B,A) )
           => ( r2_hidden(A,k1_relat_1(k2_funct_6(B)))
              & k1_funct_1(k2_funct_6(B),A) = k1_relat_1(C)
              & r2_hidden(A,k1_relat_1(k3_funct_6(B)))
              & k1_funct_1(k3_funct_6(B),A) = k2_relat_1(C) ) ) ) ) )).

fof(t32_funct_6,axiom,
    ( k2_funct_6(k1_xboole_0) = k1_xboole_0
    & k3_funct_6(k1_xboole_0) = k1_xboole_0 )).

fof(t33_funct_6,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( k2_funct_6(k9_finseq_1(A)) = k9_finseq_1(k1_relat_1(A))
        & k3_funct_6(k9_finseq_1(A)) = k9_finseq_1(k2_relat_1(A)) ) ) )).

fof(t34_funct_6,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( k2_funct_6(k10_finseq_1(A,B)) = k10_finseq_1(k1_relat_1(A),k1_relat_1(B))
            & k3_funct_6(k10_finseq_1(A,B)) = k10_finseq_1(k2_relat_1(A),k2_relat_1(B)) ) ) ) )).

fof(t35_funct_6,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ! [C] :
              ( ( v1_relat_1(C)
                & v1_funct_1(C) )
             => ( k2_funct_6(k11_finseq_1(A,B,C)) = k11_finseq_1(k1_relat_1(A),k1_relat_1(B),k1_relat_1(C))
                & k3_funct_6(k11_finseq_1(A,B,C)) = k11_finseq_1(k2_relat_1(A),k2_relat_1(B),k2_relat_1(C)) ) ) ) ) )).

fof(t36_funct_6,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( k2_funct_6(k2_funcop_1(A,B)) = k2_funcop_1(A,k1_relat_1(B))
        & k3_funct_6(k2_funcop_1(A,B)) = k2_funcop_1(A,k2_relat_1(B)) ) ) )).

fof(t37_funct_6,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( B != k1_xboole_0
       => ( r2_hidden(A,k4_funct_6(B))
        <=> ! [C] :
              ( r2_hidden(C,k1_relat_1(B))
             => r2_hidden(A,k1_funct_1(B,C)) ) ) ) ) )).

fof(t38_funct_6,axiom,(
    k4_funct_6(k1_xboole_0) = k1_xboole_0 )).

fof(t39_funct_6,axiom,(
    ! [A] :
      ( k3_card_3(k9_finseq_1(A)) = A
      & k4_funct_6(k9_finseq_1(A)) = A ) )).

fof(t40_funct_6,axiom,(
    ! [A,B] :
      ( k3_card_3(k10_finseq_1(A,B)) = k2_xboole_0(A,B)
      & k4_funct_6(k10_finseq_1(A,B)) = k3_xboole_0(A,B) ) )).

fof(t41_funct_6,axiom,(
    ! [A,B,C] :
      ( k3_card_3(k11_finseq_1(A,B,C)) = k2_xboole_0(k2_xboole_0(A,B),C)
      & k4_funct_6(k11_finseq_1(A,B,C)) = k3_xboole_0(k3_xboole_0(A,B),C) ) )).

fof(t42_funct_6,axiom,(
    ! [A] :
      ( k3_card_3(k2_funcop_1(k1_xboole_0,A)) = k1_xboole_0
      & k4_funct_6(k2_funcop_1(k1_xboole_0,A)) = k1_xboole_0 ) )).

fof(t43_funct_6,axiom,(
    ! [A,B] :
      ( A != k1_xboole_0
     => ( k3_card_3(k2_funcop_1(A,B)) = B
        & k4_funct_6(k2_funcop_1(A,B)) = B ) ) )).

fof(d5_funct_6,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B,C] : k5_funct_6(A,B,C) = k1_funct_1(k4_funct_5(A),k4_tarski(B,C)) ) )).

fof(t44_funct_6,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ! [D] :
          ( ( v1_relat_1(D)
            & v1_funct_1(D) )
         => ( ( r2_hidden(A,k1_relat_1(C))
              & D = k1_funct_1(C,A)
              & r2_hidden(B,k1_relat_1(D)) )
           => k5_funct_6(C,A,B) = k1_funct_1(D,B) ) ) ) )).

fof(t45_funct_6,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ! [C] :
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => ! [D] :
              ( ( v1_relat_1(D)
                & v1_funct_1(D) )
             => ( r2_hidden(A,k1_relat_1(B))
               => ( k5_funct_6(k9_finseq_1(B),np__1,A) = k1_funct_1(B,A)
                  & k5_funct_6(k10_finseq_1(B,C),np__1,A) = k1_funct_1(B,A)
                  & k5_funct_6(k11_finseq_1(B,C,D),np__1,A) = k1_funct_1(B,A) ) ) ) ) ) )).

fof(t46_funct_6,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ! [C] :
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => ! [D] :
              ( ( v1_relat_1(D)
                & v1_funct_1(D) )
             => ( r2_hidden(A,k1_relat_1(B))
               => ( k5_funct_6(k10_finseq_1(C,B),np__2,A) = k1_funct_1(B,A)
                  & k5_funct_6(k11_finseq_1(C,B,D),np__2,A) = k1_funct_1(B,A) ) ) ) ) ) )).

fof(t47_funct_6,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ! [C] :
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => ! [D] :
              ( ( v1_relat_1(D)
                & v1_funct_1(D) )
             => ( r2_hidden(A,k1_relat_1(B))
               => k5_funct_6(k11_finseq_1(C,D,B),np__3,A) = k1_funct_1(B,A) ) ) ) ) )).

fof(t48_funct_6,axiom,(
    ! [A,B,C,D] :
      ( ( v1_relat_1(D)
        & v1_funct_1(D) )
     => ( ( r2_hidden(A,B)
          & r2_hidden(C,k1_relat_1(D)) )
       => k5_funct_6(k2_funcop_1(B,D),A,C) = k1_funct_1(D,C) ) ) )).

fof(d6_funct_6,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => k6_funct_6(A) = k3_funct_5(k7_relat_1(k6_funct_5(A),k2_zfmisc_1(k4_funct_6(k2_funct_6(A)),k1_relat_1(A)))) ) )).

fof(t49_funct_6,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( k1_relat_1(k6_funct_6(A)) = k4_funct_6(k2_funct_6(A))
        & r1_tarski(k2_relat_1(k6_funct_6(A)),k4_card_3(k3_funct_6(A))) ) ) )).

fof(t50_funct_6,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( r2_hidden(A,k1_relat_1(k6_funct_6(B)))
       => ( v1_relat_1(k1_funct_1(k6_funct_6(B),A))
          & v1_funct_1(k1_funct_1(k6_funct_6(B),A)) ) ) ) )).

fof(t51_funct_6,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ! [C] :
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => ( ( r2_hidden(A,k1_relat_1(k6_funct_6(B)))
              & C = k1_funct_1(k6_funct_6(B),A) )
           => ( k1_relat_1(C) = k10_relat_1(B,k1_funct_6(k2_relat_1(B)))
              & ! [D] :
                  ( r2_hidden(D,k1_relat_1(C))
                 => ( r2_hidden(k4_tarski(D,A),k1_relat_1(k4_funct_5(B)))
                    & k1_funct_1(C,D) = k1_funct_1(k4_funct_5(B),k4_tarski(D,A)) ) ) ) ) ) ) )).

fof(t52_funct_6,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( r2_hidden(A,k1_relat_1(k6_funct_6(B)))
       => ! [C] :
            ( ( v1_relat_1(C)
              & v1_funct_1(C) )
           => ( r2_hidden(C,k2_relat_1(B))
             => r2_hidden(A,k1_relat_1(C)) ) ) ) ) )).

fof(t53_funct_6,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ! [C] :
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => ( ( r2_hidden(B,k2_relat_1(C))
              & ! [D] :
                  ( ( v1_relat_1(D)
                    & v1_funct_1(D) )
                 => ( r2_hidden(D,k2_relat_1(C))
                   => r2_hidden(A,k1_relat_1(D)) ) ) )
           => r2_hidden(A,k1_relat_1(k6_funct_6(C))) ) ) ) )).

fof(t54_funct_6,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ! [D] :
          ( ( v1_relat_1(D)
            & v1_funct_1(D) )
         => ! [E] :
              ( ( v1_relat_1(E)
                & v1_funct_1(E) )
             => ( ( r2_hidden(A,k1_relat_1(C))
                  & D = k1_funct_1(C,A)
                  & r2_hidden(B,k1_relat_1(k6_funct_6(C)))
                  & E = k1_funct_1(k6_funct_6(C),B) )
               => k1_funct_1(D,B) = k1_funct_1(E,A) ) ) ) ) )).

fof(t55_funct_6,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ( ( r2_hidden(A,k1_relat_1(C))
          & v1_relat_1(k1_funct_1(C,A))
          & v1_funct_1(k1_funct_1(C,A))
          & r2_hidden(B,k1_relat_1(k6_funct_6(C))) )
       => k5_funct_6(C,A,B) = k5_funct_6(k6_funct_6(C),B,A) ) ) )).

fof(d7_funct_6,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( B = k7_funct_6(A)
          <=> ( k1_relat_1(B) = k4_card_3(k2_funct_6(A))
              & ! [C] :
                  ( ( v1_relat_1(C)
                    & v1_funct_1(C) )
                 => ~ ( r2_hidden(C,k4_card_3(k2_funct_6(A)))
                      & ! [D] :
                          ( ( v1_relat_1(D)
                            & v1_funct_1(D) )
                         => ~ ( k1_funct_1(B,C) = D
                              & k1_relat_1(D) = k10_relat_1(A,k1_funct_6(k2_relat_1(A)))
                              & ! [E] :
                                  ( r2_hidden(E,k1_relat_1(D))
                                 => k1_funct_1(D,E) = k1_funct_1(k4_funct_5(A),k4_tarski(E,k1_funct_1(C,E))) ) ) ) ) ) ) ) ) ) )).

fof(t56_funct_6,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ! [C] :
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => ( ( r2_hidden(B,k4_card_3(k2_funct_6(C)))
              & r2_hidden(A,k1_relat_1(B)) )
           => k5_funct_6(k7_funct_6(C),B,A) = k5_funct_6(C,A,k1_funct_1(B,A)) ) ) ) )).

fof(t57_funct_6,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ! [C] :
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => ! [D] :
              ( ( v1_relat_1(D)
                & v1_funct_1(D) )
             => ! [E] :
                  ( ( v1_relat_1(E)
                    & v1_funct_1(E) )
                 => ( ( r2_hidden(A,k1_relat_1(B))
                      & C = k1_funct_1(B,A)
                      & r2_hidden(D,k4_card_3(k2_funct_6(B)))
                      & E = k1_funct_1(k7_funct_6(B),D) )
                   => ( r2_hidden(k1_funct_1(D,A),k1_relat_1(C))
                      & k1_funct_1(E,A) = k1_funct_1(C,k1_funct_1(D,A))
                      & r2_hidden(E,k4_card_3(k3_funct_6(B))) ) ) ) ) ) ) )).

fof(t58_funct_6,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => k2_relat_1(k7_funct_6(A)) = k4_card_3(k3_funct_6(A)) ) )).

fof(t59_funct_6,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( ~ r2_hidden(k1_xboole_0,k2_relat_1(A))
       => ( v2_funct_1(k7_funct_6(A))
        <=> ! [B] :
              ( ( v1_relat_1(B)
                & v1_funct_1(B) )
             => ( r2_hidden(B,k2_relat_1(A))
               => v2_funct_1(B) ) ) ) ) ) )).

fof(t60_funct_6,axiom,
    ( k6_funct_6(k1_xboole_0) = k1_xboole_0
    & k7_funct_6(k1_xboole_0) = k2_funcop_1(k1_tarski(k1_xboole_0),k1_xboole_0) )).

fof(t61_funct_6,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( k1_relat_1(k6_funct_6(k9_finseq_1(A))) = k1_relat_1(A)
        & ! [B] :
            ( r2_hidden(B,k1_relat_1(A))
           => k1_funct_1(k6_funct_6(k9_finseq_1(A)),B) = k9_finseq_1(k1_funct_1(A,B)) ) ) ) )).

fof(t62_funct_6,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( k1_relat_1(k6_funct_6(k10_finseq_1(A,B))) = k3_xboole_0(k1_relat_1(A),k1_relat_1(B))
            & ! [C] :
                ( r2_hidden(C,k3_xboole_0(k1_relat_1(A),k1_relat_1(B)))
               => k1_funct_1(k6_funct_6(k10_finseq_1(A,B)),C) = k10_finseq_1(k1_funct_1(A,C),k1_funct_1(B,C)) ) ) ) ) )).

fof(t63_funct_6,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( A != k1_xboole_0
       => ( k1_relat_1(k6_funct_6(k2_funcop_1(A,B))) = k1_relat_1(B)
          & ! [C] :
              ( r2_hidden(C,k1_relat_1(B))
             => k1_funct_1(k6_funct_6(k2_funcop_1(A,B)),C) = k2_funcop_1(A,k1_funct_1(B,C)) ) ) ) ) )).

fof(t64_funct_6,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( k1_relat_1(k7_funct_6(k9_finseq_1(A))) = k4_card_3(k9_finseq_1(k1_relat_1(A)))
        & k2_relat_1(k7_funct_6(k9_finseq_1(A))) = k4_card_3(k9_finseq_1(k2_relat_1(A)))
        & ! [B] :
            ( r2_hidden(B,k1_relat_1(A))
           => k1_funct_1(k7_funct_6(k9_finseq_1(A)),k9_finseq_1(B)) = k9_finseq_1(k1_funct_1(A,B)) ) ) ) )).

fof(t65_funct_6,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( k1_relat_1(k7_funct_6(k10_finseq_1(A,B))) = k4_card_3(k10_finseq_1(k1_relat_1(A),k1_relat_1(B)))
            & k2_relat_1(k7_funct_6(k10_finseq_1(A,B))) = k4_card_3(k10_finseq_1(k2_relat_1(A),k2_relat_1(B)))
            & ! [C,D] :
                ( ( r2_hidden(C,k1_relat_1(A))
                  & r2_hidden(D,k1_relat_1(B)) )
               => k1_funct_1(k7_funct_6(k10_finseq_1(A,B)),k10_finseq_1(C,D)) = k10_finseq_1(k1_funct_1(A,C),k1_funct_1(B,D)) ) ) ) ) )).

fof(t66_funct_6,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( k1_relat_1(k7_funct_6(k2_funcop_1(A,B))) = k1_funct_2(A,k1_relat_1(B))
        & k2_relat_1(k7_funct_6(k2_funcop_1(A,B))) = k1_funct_2(A,k2_relat_1(B))
        & ! [C] :
            ( ( v1_relat_1(C)
              & v1_funct_1(C) )
           => ( r2_hidden(C,k1_funct_2(A,k1_relat_1(B)))
             => k1_funct_1(k7_funct_6(k2_funcop_1(A,B)),C) = k5_relat_1(C,B) ) ) ) ) )).

fof(t67_funct_6,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ! [C] :
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => ( ( r2_hidden(A,k1_relat_1(B))
              & r2_hidden(A,k1_relat_1(C)) )
           => ! [D,E] :
                ( k1_funct_1(k13_funct_3(B,C),A) = k4_tarski(D,E)
              <=> k1_funct_1(k6_funct_6(k10_finseq_1(B,C)),A) = k10_finseq_1(D,E) ) ) ) ) )).

fof(t68_funct_6,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ! [D] :
          ( ( v1_relat_1(D)
            & v1_funct_1(D) )
         => ( ( r2_hidden(A,k1_relat_1(C))
              & r2_hidden(B,k1_relat_1(D)) )
           => ! [E,F] :
                ( k1_funct_1(k15_funct_3(C,D),k4_tarski(A,B)) = k4_tarski(E,F)
              <=> k1_funct_1(k7_funct_6(k10_finseq_1(C,D)),k10_finseq_1(A,B)) = k10_finseq_1(E,F) ) ) ) ) )).

fof(t69_funct_6,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ! [C] :
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => ( ( k1_relat_1(B) = A
              & k1_relat_1(C) = A
              & ! [D] :
                  ( r2_hidden(D,A)
                 => r2_wellord2(k1_funct_1(B,D),k1_funct_1(C,D)) ) )
           => r2_wellord2(k4_card_3(B),k4_card_3(C)) ) ) ) )).

fof(t70_funct_6,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ! [C] :
              ( ( v1_relat_1(C)
                & v1_funct_1(C) )
             => ( ( k1_relat_1(A) = k1_relat_1(B)
                  & k1_relat_1(C) = k2_relat_1(B)
                  & v2_funct_1(B)
                  & ! [D] :
                      ( r2_hidden(D,k1_relat_1(B))
                     => r2_wellord2(k1_funct_1(A,D),k1_funct_1(C,k1_funct_1(B,D))) ) )
               => r2_wellord2(k4_card_3(A),k4_card_3(C)) ) ) ) ) )).

fof(t71_funct_6,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,A,A)
            & v3_funct_2(C,A,A)
            & m2_relset_1(C,A,A) )
         => ( k1_relat_1(B) = A
           => r2_wellord2(k4_card_3(B),k4_card_3(k5_relat_1(C,B))) ) ) ) )).

fof(d8_funct_6,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B,C] :
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => ( C = k8_funct_6(A,B)
          <=> ( k1_relat_1(C) = k1_relat_1(A)
              & ! [D] :
                  ( r2_hidden(D,k1_relat_1(A))
                 => k1_funct_1(C,D) = k1_funct_2(k1_funct_1(A,D),B) ) ) ) ) ) )).

fof(t72_funct_6,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( ~ r2_hidden(k1_xboole_0,k2_relat_1(A))
       => k8_funct_6(A,k1_xboole_0) = k2_funcop_1(k1_relat_1(A),k1_xboole_0) ) ) )).

fof(t73_funct_6,axiom,(
    ! [A] : k8_funct_6(k1_xboole_0,A) = k1_xboole_0 )).

fof(t74_funct_6,axiom,(
    ! [A,B] : k8_funct_6(k9_finseq_1(A),B) = k9_finseq_1(k1_funct_2(A,B)) )).

fof(t75_funct_6,axiom,(
    ! [A,B,C] : k8_funct_6(k10_finseq_1(A,B),C) = k10_finseq_1(k1_funct_2(A,C),k1_funct_2(B,C)) )).

fof(t76_funct_6,axiom,(
    ! [A,B,C] : k8_funct_6(k2_funcop_1(A,B),C) = k2_funcop_1(A,k1_funct_2(B,C)) )).

fof(t77_funct_6,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => r2_wellord2(k1_funct_2(k3_card_3(k2_card_3(B)),A),k4_card_3(k8_funct_6(B,A))) ) )).

fof(d9_funct_6,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ! [C] :
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => ( C = k9_funct_6(A,B)
          <=> ( k1_relat_1(C) = k1_relat_1(B)
              & ! [D] :
                  ( r2_hidden(D,k1_relat_1(B))
                 => k1_funct_1(C,D) = k1_funct_2(A,k1_funct_1(B,D)) ) ) ) ) ) )).

fof(t78_funct_6,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => k9_funct_6(k1_xboole_0,A) = k2_funcop_1(k1_relat_1(A),k1_tarski(k1_xboole_0)) ) )).

fof(t79_funct_6,axiom,(
    ! [A] : k9_funct_6(A,k1_xboole_0) = k1_xboole_0 )).

fof(t80_funct_6,axiom,(
    ! [A,B] : k9_funct_6(A,k9_finseq_1(B)) = k9_finseq_1(k1_funct_2(A,B)) )).

fof(t81_funct_6,axiom,(
    ! [A,B,C] : k9_funct_6(A,k10_finseq_1(B,C)) = k10_finseq_1(k1_funct_2(A,B),k1_funct_2(A,C)) )).

fof(t82_funct_6,axiom,(
    ! [A,B,C] : k9_funct_6(A,k2_funcop_1(B,C)) = k2_funcop_1(B,k1_funct_2(A,C)) )).

fof(t83_funct_6,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => r2_wellord2(k4_card_3(k9_funct_6(A,B)),k1_funct_2(A,k4_card_3(B))) ) )).

fof(d10_funct_6,axiom,(
    $true )).

fof(d11_funct_6,axiom,(
    $true )).

fof(d12_funct_6,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => k10_funct_6(A) = k5_funct_5(k4_funct_5(A)) ) )).

fof(t84_funct_6,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( r2_hidden(B,k1_relat_1(k10_funct_6(A)))
         => ( v1_relat_1(k1_funct_1(k10_funct_6(A),B))
            & v1_funct_1(k1_funct_1(k10_funct_6(A),B)) ) ) ) )).

fof(t85_funct_6,axiom,(
    ! [A,B,C,D] :
      ( ( v1_relat_1(D)
        & v1_funct_1(D) )
     => ( r2_hidden(D,k1_funct_2(A,k1_funct_2(B,C)))
       => ( A = k1_xboole_0
          | B = k1_xboole_0
          | r2_hidden(k10_funct_6(D),k1_funct_2(B,k1_funct_2(A,C))) ) ) ) )).

fof(t86_funct_6,axiom,(
    ! [A,B,C,D] :
      ( ( v1_relat_1(D)
        & v1_funct_1(D) )
     => ( r2_hidden(D,k1_funct_2(A,k1_funct_2(B,C)))
       => ( A = k1_xboole_0
          | B = k1_xboole_0
          | ! [E] :
              ( ( v1_relat_1(E)
                & v1_funct_1(E) )
             => ! [F] :
                  ( ( v1_relat_1(F)
                    & v1_funct_1(F) )
                 => ! [G,H] :
                      ( ( r2_hidden(G,A)
                        & r2_hidden(H,B)
                        & k1_funct_1(D,G) = E
                        & k1_funct_1(k10_funct_6(D),H) = F )
                     => ( k1_funct_1(F,G) = k1_funct_1(E,H)
                        & k1_relat_1(F) = A
                        & k1_relat_1(E) = B
                        & r1_tarski(k2_relat_1(F),C)
                        & r1_tarski(k2_relat_1(E),C) ) ) ) ) ) ) ) )).

fof(t87_funct_6,axiom,(
    ! [A,B,C,D] :
      ( ( v1_relat_1(D)
        & v1_funct_1(D) )
     => ( r2_hidden(D,k1_funct_2(A,k1_funct_2(B,C)))
       => ( A = k1_xboole_0
          | B = k1_xboole_0
          | k10_funct_6(k10_funct_6(D)) = D ) ) ) )).

fof(t88_funct_6,axiom,(
    k10_funct_6(k1_xboole_0) = k1_xboole_0 )).

fof(t89_funct_6,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_funcop_1(A) )
     => k1_relat_1(k2_funct_6(A)) = k1_relat_1(A) ) )).

fof(t90_funct_6,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_funcop_1(A) )
     => k1_relat_1(k3_funct_6(A)) = k1_relat_1(A) ) )).

fof(dt_k1_funct_6,axiom,(
    $true )).

fof(dt_k2_funct_6,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v1_relat_1(k2_funct_6(A))
        & v1_funct_1(k2_funct_6(A)) ) ) )).

fof(dt_k3_funct_6,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v1_relat_1(k3_funct_6(A))
        & v1_funct_1(k3_funct_6(A)) ) ) )).

fof(dt_k4_funct_6,axiom,(
    $true )).

fof(dt_k5_funct_6,axiom,(
    $true )).

fof(dt_k6_funct_6,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v1_relat_1(k6_funct_6(A))
        & v1_funct_1(k6_funct_6(A)) ) ) )).

fof(dt_k7_funct_6,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v1_relat_1(k7_funct_6(A))
        & v1_funct_1(k7_funct_6(A)) ) ) )).

fof(dt_k8_funct_6,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v1_relat_1(k8_funct_6(A,B))
        & v1_funct_1(k8_funct_6(A,B)) ) ) )).

fof(dt_k9_funct_6,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( v1_relat_1(k9_funct_6(A,B))
        & v1_funct_1(k9_funct_6(A,B)) ) ) )).

fof(dt_k10_funct_6,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v1_relat_1(k10_funct_6(A))
        & v1_funct_1(k10_funct_6(A))
        & v1_funcop_1(k10_funct_6(A)) ) ) )).

fof(t5_funct_6,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => k4_card_3(k10_finseq_1(A,B)) = a_2_0_funct_6(A,B) ) ) )).

fof(t7_funct_6,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => k4_card_3(k11_finseq_1(A,B,C)) = a_3_0_funct_6(A,B,C) ) ) ) )).

fof(fraenkel_a_2_0_funct_6,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C) )
     => ( r2_hidden(A,a_2_0_funct_6(B,C))
      <=> ? [D,E] :
            ( m1_subset_1(D,B)
            & m1_subset_1(E,C)
            & A = k10_finseq_1(D,E) ) ) ) )).

fof(fraenkel_a_3_0_funct_6,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & ~ v1_xboole_0(D) )
     => ( r2_hidden(A,a_3_0_funct_6(B,C,D))
      <=> ? [E,F,G] :
            ( m1_subset_1(E,B)
            & m1_subset_1(F,C)
            & m1_subset_1(G,D)
            & A = k11_finseq_1(E,F,G) ) ) ) )).
%------------------------------------------------------------------------------