SET007 Axioms: SET007+25.ax


%------------------------------------------------------------------------------
% File     : SET007+25 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : The Modification of a Function by a Function
% Version  : [Urb08] axioms.
% English  : and the Iteration of the Composition of a Function

% 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_4 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   95 (   6 unit)
%            Number of atoms       :  501 (  86 equality)
%            Maximal formula depth :   17 (   7 average)
%            Number of connectives :  428 (  22 ~  ;   3  |; 204  &)
%                                         (  12 <=>; 187 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   13 (   1 propositional; 0-3 arity)
%            Number of functors    :   20 (   1 constant; 0-5 arity)
%            Number of variables   :  317 (   0 singleton; 309 !;   8 ?)
%            Maximal term depth    :    4 (   1 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(fc1_funct_4,axiom,(
    ! [A,B,C,D] :
      ( v1_relat_1(k4_funct_4(A,B,C,D))
      & v1_funct_1(k4_funct_4(A,B,C,D)) ) )).

fof(t1_funct_4,axiom,(
    ! [A] :
      ~ ( ! [B] :
            ~ ( r2_hidden(B,A)
              & ! [C,D] : B != k4_tarski(C,D) )
        & ! [B,C] : ~ r1_tarski(A,k2_zfmisc_1(B,C)) ) )).

fof(t2_funct_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => k5_relat_1(B,A) = k5_relat_1(B,k7_relat_1(A,k2_relat_1(B))) ) ) )).

fof(t3_funct_4,axiom,(
    $true )).

fof(t4_funct_4,axiom,(
    ! [A,B] :
      ( r1_tarski(k6_partfun1(A),k6_partfun1(B))
    <=> r1_tarski(A,B) ) )).

fof(t5_funct_4,axiom,(
    ! [A,B,C] :
      ( r1_tarski(A,B)
     => r1_tarski(k2_funcop_1(A,C),k2_funcop_1(B,C)) ) )).

fof(t6_funct_4,axiom,(
    ! [A,B,C,D] :
      ( r1_tarski(k2_funcop_1(A,B),k2_funcop_1(C,D))
     => r1_tarski(A,C) ) )).

fof(t7_funct_4,axiom,(
    ! [A,B,C,D] :
      ( r1_tarski(k2_funcop_1(A,B),k2_funcop_1(C,D))
     => ( A = k1_xboole_0
        | B = D ) ) )).

fof(t8_funct_4,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( r2_hidden(A,k1_relat_1(B))
       => r1_tarski(k2_funcop_1(k1_tarski(A),k1_funct_1(B,A)),B) ) ) )).

fof(t9_funct_4,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => r1_tarski(k7_relat_1(k8_relat_1(A,C),B),C) ) )).

fof(t10_funct_4,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ! [D] :
          ( ( v1_relat_1(D)
            & v1_funct_1(D) )
         => ( r1_tarski(C,D)
           => r1_tarski(k7_relat_1(k8_relat_1(A,C),B),k7_relat_1(k8_relat_1(A,D),B)) ) ) ) )).

fof(d1_funct_4,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) )
             => ( C = k1_funct_4(A,B)
              <=> ( k1_relat_1(C) = k2_xboole_0(k1_relat_1(A),k1_relat_1(B))
                  & ! [D] :
                      ( r2_hidden(D,k2_xboole_0(k1_relat_1(A),k1_relat_1(B)))
                     => ( ( r2_hidden(D,k1_relat_1(B))
                         => k1_funct_1(C,D) = k1_funct_1(B,D) )
                        & ( ~ r2_hidden(D,k1_relat_1(B))
                         => k1_funct_1(C,D) = k1_funct_1(A,D) ) ) ) ) ) ) ) ) )).

fof(t11_funct_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( r1_tarski(k1_relat_1(A),k1_relat_1(k1_funct_4(A,B)))
            & r1_tarski(k1_relat_1(B),k1_relat_1(k1_funct_4(A,B))) ) ) ) )).

fof(t12_funct_4,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))
           => k1_funct_1(k1_funct_4(C,B),A) = k1_funct_1(C,A) ) ) ) )).

fof(t13_funct_4,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(k1_funct_4(B,C)))
          <=> ( r2_hidden(A,k1_relat_1(B))
              | r2_hidden(A,k1_relat_1(C)) ) ) ) ) )).

fof(t14_funct_4,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))
           => k1_funct_1(k1_funct_4(C,B),A) = k1_funct_1(B,A) ) ) ) )).

fof(t15_funct_4,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_4(k1_funct_4(A,B),C) = k1_funct_4(A,k1_funct_4(B,C)) ) ) ) )).

fof(t16_funct_4,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ! [C] :
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => ( ( r1_partfun1(B,C)
              & r2_hidden(A,k1_relat_1(B)) )
           => k1_funct_1(k1_funct_4(B,C),A) = k1_funct_1(B,A) ) ) ) )).

fof(t17_funct_4,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ! [C] :
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => ( ( r1_xboole_0(k1_relat_1(B),k1_relat_1(C))
              & r2_hidden(A,k1_relat_1(B)) )
           => k1_funct_1(k1_funct_4(B,C),A) = k1_funct_1(B,A) ) ) ) )).

fof(t18_funct_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => r1_tarski(k2_relat_1(k1_funct_4(A,B)),k2_xboole_0(k2_relat_1(A),k2_relat_1(B))) ) ) )).

fof(t19_funct_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => r1_tarski(k2_relat_1(A),k2_relat_1(k1_funct_4(B,A))) ) ) )).

fof(t20_funct_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( r1_tarski(k1_relat_1(A),k1_relat_1(B))
           => k1_funct_4(A,B) = B ) ) ) )).

fof(t21_funct_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => k1_funct_4(k1_xboole_0,A) = A ) )).

fof(t22_funct_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => k1_funct_4(A,k1_xboole_0) = A ) )).

fof(t23_funct_4,axiom,(
    ! [A,B] : k1_funct_4(k6_partfun1(A),k6_partfun1(B)) = k6_partfun1(k2_xboole_0(A,B)) )).

fof(t24_funct_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => k7_relat_1(k1_funct_4(A,B),k1_relat_1(B)) = B ) ) )).

fof(t25_funct_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => r1_tarski(k7_relat_1(k1_funct_4(A,B),k4_xboole_0(k1_relat_1(A),k1_relat_1(B))),A) ) ) )).

fof(t26_funct_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => r1_tarski(A,k1_funct_4(B,A)) ) ) )).

fof(t27_funct_4,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) )
             => ( r1_partfun1(A,k1_funct_4(B,C))
               => r1_partfun1(k7_relat_1(A,k4_xboole_0(k1_relat_1(A),k1_relat_1(C))),B) ) ) ) ) )).

fof(t28_funct_4,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) )
             => ( r1_partfun1(A,k1_funct_4(B,C))
               => r1_partfun1(A,C) ) ) ) ) )).

fof(t29_funct_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( r1_partfun1(A,B)
          <=> r1_tarski(A,k1_funct_4(A,B)) ) ) ) )).

fof(t30_funct_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => r1_tarski(k1_funct_4(A,B),k2_xboole_0(A,B)) ) ) )).

fof(t31_funct_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( r1_partfun1(A,B)
          <=> k2_xboole_0(A,B) = k1_funct_4(A,B) ) ) ) )).

fof(t32_funct_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( r1_xboole_0(k1_relat_1(A),k1_relat_1(B))
           => k2_xboole_0(A,B) = k1_funct_4(A,B) ) ) ) )).

fof(t33_funct_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( r1_xboole_0(k1_relat_1(A),k1_relat_1(B))
           => r1_tarski(A,k1_funct_4(A,B)) ) ) ) )).

fof(t34_funct_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( r1_xboole_0(k1_relat_1(A),k1_relat_1(B))
           => k7_relat_1(k1_funct_4(A,B),k1_relat_1(A)) = A ) ) ) )).

fof(t35_funct_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( r1_partfun1(A,B)
          <=> k1_funct_4(A,B) = k1_funct_4(B,A) ) ) ) )).

fof(t36_funct_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( r1_xboole_0(k1_relat_1(A),k1_relat_1(B))
           => k1_funct_4(A,B) = k1_funct_4(B,A) ) ) ) )).

fof(t37_funct_4,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(C)
        & m2_relset_1(C,A,B) )
     => ! [D] :
          ( ( v1_funct_1(D)
            & m2_relset_1(D,A,B) )
         => ( v1_partfun1(D,A,B)
           => k1_funct_4(C,D) = D ) ) ) )).

fof(t38_funct_4,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(C)
        & v1_funct_2(C,A,B)
        & m2_relset_1(C,A,B) )
     => ! [D] :
          ( ( v1_funct_1(D)
            & v1_funct_2(D,A,B)
            & m2_relset_1(D,A,B) )
         => ( ( B = k1_xboole_0
             => A = k1_xboole_0 )
           => k1_funct_4(C,D) = D ) ) ) )).

fof(t39_funct_4,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,A,A)
        & m2_relset_1(B,A,A) )
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,A,A)
            & m2_relset_1(C,A,A) )
         => k1_funct_4(B,C) = C ) ) )).

fof(t40_funct_4,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,A,B)
            & m2_relset_1(C,A,B) )
         => ! [D] :
              ( ( v1_funct_1(D)
                & v1_funct_2(D,A,B)
                & m2_relset_1(D,A,B) )
             => k1_funct_4(C,D) = D ) ) ) )).

fof(t41_funct_4,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(C)
        & m2_relset_1(C,A,B) )
     => ! [D] :
          ( ( v1_funct_1(D)
            & m2_relset_1(D,A,B) )
         => ( v1_funct_1(k1_funct_4(C,D))
            & m2_relset_1(k1_funct_4(C,D),A,B) ) ) ) )).

fof(d2_funct_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( B = k2_funct_4(A)
          <=> ( ! [C] :
                  ( r2_hidden(C,k1_relat_1(B))
                <=> ? [D,E] :
                      ( C = k4_tarski(E,D)
                      & r2_hidden(k4_tarski(D,E),k1_relat_1(A)) ) )
              & ! [C,D] :
                  ( r2_hidden(k4_tarski(C,D),k1_relat_1(A))
                 => k1_funct_1(B,k4_tarski(D,C)) = k1_funct_1(A,k4_tarski(C,D)) ) ) ) ) ) )).

fof(t42_funct_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => r1_tarski(k2_relat_1(k2_funct_4(A)),k2_relat_1(A)) ) )).

fof(t43_funct_4,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ( r2_hidden(k4_tarski(A,B),k1_relat_1(C))
      <=> r2_hidden(k4_tarski(B,A),k1_relat_1(k2_funct_4(C))) ) ) )).

fof(t44_funct_4,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ( r2_hidden(k4_tarski(A,B),k1_relat_1(k2_funct_4(C)))
       => k1_funct_1(k2_funct_4(C),k4_tarski(A,B)) = k1_funct_1(C,k4_tarski(B,A)) ) ) )).

fof(t45_funct_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ? [B,C] : r1_tarski(k1_relat_1(k2_funct_4(A)),k2_zfmisc_1(B,C)) ) )).

fof(t46_funct_4,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ( r1_tarski(k1_relat_1(C),k2_zfmisc_1(A,B))
       => r1_tarski(k1_relat_1(k2_funct_4(C)),k2_zfmisc_1(B,A)) ) ) )).

fof(t47_funct_4,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ( k1_relat_1(C) = k2_zfmisc_1(A,B)
       => k1_relat_1(k2_funct_4(C)) = k2_zfmisc_1(B,A) ) ) )).

fof(t48_funct_4,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ( r1_tarski(k1_relat_1(C),k2_zfmisc_1(A,B))
       => k2_relat_1(k2_funct_4(C)) = k2_relat_1(C) ) ) )).

fof(t49_funct_4,axiom,(
    ! [A,B,C,D] :
      ( ( v1_funct_1(D)
        & m2_relset_1(D,k2_zfmisc_1(A,B),C) )
     => ( v1_funct_1(k2_funct_4(D))
        & m2_relset_1(k2_funct_4(D),k2_zfmisc_1(B,A),C) ) ) )).

fof(t50_funct_4,axiom,(
    ! [A,B,C,D] :
      ( ( v1_funct_1(D)
        & v1_funct_2(D,k2_zfmisc_1(A,B),C)
        & m2_relset_1(D,k2_zfmisc_1(A,B),C) )
     => ( C != k1_xboole_0
       => ( v1_funct_1(k2_funct_4(D))
          & v1_funct_2(k2_funct_4(D),k2_zfmisc_1(B,A),C)
          & m2_relset_1(k2_funct_4(D),k2_zfmisc_1(B,A),C) ) ) ) )).

fof(t51_funct_4,axiom,(
    ! [A,B,C] :
      ( ~ v1_xboole_0(C)
     => ! [D] :
          ( ( v1_funct_1(D)
            & v1_funct_2(D,k2_zfmisc_1(A,B),C)
            & m2_relset_1(D,k2_zfmisc_1(A,B),C) )
         => ( v1_funct_1(k2_funct_4(D))
            & v1_funct_2(k2_funct_4(D),k2_zfmisc_1(B,A),C)
            & m2_relset_1(k2_funct_4(D),k2_zfmisc_1(B,A),C) ) ) ) )).

fof(t52_funct_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => r1_tarski(k2_funct_4(k2_funct_4(A)),A) ) )).

fof(t53_funct_4,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ( r1_tarski(k1_relat_1(C),k2_zfmisc_1(A,B))
       => k2_funct_4(k2_funct_4(C)) = C ) ) )).

fof(t54_funct_4,axiom,(
    ! [A,B,C,D] :
      ( ( v1_funct_1(D)
        & m2_relset_1(D,k2_zfmisc_1(A,B),C) )
     => k2_funct_4(k2_funct_4(D)) = D ) )).

fof(t55_funct_4,axiom,(
    ! [A,B,C,D] :
      ( ( v1_funct_1(D)
        & v1_funct_2(D,k2_zfmisc_1(A,B),C)
        & m2_relset_1(D,k2_zfmisc_1(A,B),C) )
     => ( C != k1_xboole_0
       => k2_funct_4(k2_funct_4(D)) = D ) ) )).

fof(t56_funct_4,axiom,(
    ! [A,B,C] :
      ( ~ v1_xboole_0(C)
     => ! [D] :
          ( ( v1_funct_1(D)
            & v1_funct_2(D,k2_zfmisc_1(A,B),C)
            & m2_relset_1(D,k2_zfmisc_1(A,B),C) )
         => k2_funct_4(k2_funct_4(D)) = D ) ) )).

fof(d3_funct_4,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) )
             => ( C = k3_funct_4(A,B)
              <=> ( ! [D] :
                      ( r2_hidden(D,k1_relat_1(C))
                    <=> ? [E,F,G,H] :
                          ( D = k4_tarski(k4_tarski(E,G),k4_tarski(F,H))
                          & r2_hidden(k4_tarski(E,F),k1_relat_1(A))
                          & r2_hidden(k4_tarski(G,H),k1_relat_1(B)) ) )
                  & ! [D,E,F,G] :
                      ( ( r2_hidden(k4_tarski(D,E),k1_relat_1(A))
                        & r2_hidden(k4_tarski(F,G),k1_relat_1(B)) )
                     => k1_funct_1(C,k4_tarski(k4_tarski(D,F),k4_tarski(E,G))) = k4_tarski(k1_funct_1(A,k4_tarski(D,E)),k1_funct_1(B,k4_tarski(F,G))) ) ) ) ) ) ) )).

fof(t57_funct_4,axiom,(
    ! [A,B,C,D,E] :
      ( ( v1_relat_1(E)
        & v1_funct_1(E) )
     => ! [F] :
          ( ( v1_relat_1(F)
            & v1_funct_1(F) )
         => ( r2_hidden(k4_tarski(k4_tarski(A,B),k4_tarski(C,D)),k1_relat_1(k3_funct_4(E,F)))
          <=> ( r2_hidden(k4_tarski(A,C),k1_relat_1(E))
              & r2_hidden(k4_tarski(B,D),k1_relat_1(F)) ) ) ) ) )).

fof(t58_funct_4,axiom,(
    ! [A,B,C,D,E] :
      ( ( v1_relat_1(E)
        & v1_funct_1(E) )
     => ! [F] :
          ( ( v1_relat_1(F)
            & v1_funct_1(F) )
         => ( r2_hidden(k4_tarski(k4_tarski(A,B),k4_tarski(C,D)),k1_relat_1(k3_funct_4(E,F)))
           => k1_funct_1(k3_funct_4(E,F),k4_tarski(k4_tarski(A,B),k4_tarski(C,D))) = k4_tarski(k1_funct_1(E,k4_tarski(A,C)),k1_funct_1(F,k4_tarski(B,D))) ) ) ) )).

fof(t59_funct_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => r1_tarski(k2_relat_1(k3_funct_4(A,B)),k2_zfmisc_1(k2_relat_1(A),k2_relat_1(B))) ) ) )).

fof(t60_funct_4,axiom,(
    ! [A,B,C,D,E] :
      ( ( v1_relat_1(E)
        & v1_funct_1(E) )
     => ! [F] :
          ( ( v1_relat_1(F)
            & v1_funct_1(F) )
         => ( ( r1_tarski(k1_relat_1(E),k2_zfmisc_1(A,B))
              & r1_tarski(k1_relat_1(F),k2_zfmisc_1(C,D)) )
           => r1_tarski(k1_relat_1(k3_funct_4(E,F)),k2_zfmisc_1(k2_zfmisc_1(A,C),k2_zfmisc_1(B,D))) ) ) ) )).

fof(t61_funct_4,axiom,(
    ! [A,B,C,D,E] :
      ( ( v1_relat_1(E)
        & v1_funct_1(E) )
     => ! [F] :
          ( ( v1_relat_1(F)
            & v1_funct_1(F) )
         => ( ( k1_relat_1(E) = k2_zfmisc_1(A,B)
              & k1_relat_1(F) = k2_zfmisc_1(C,D) )
           => k1_relat_1(k3_funct_4(E,F)) = k2_zfmisc_1(k2_zfmisc_1(A,C),k2_zfmisc_1(B,D)) ) ) ) )).

fof(t62_funct_4,axiom,(
    ! [A,B,C,D,E,F,G] :
      ( ( v1_funct_1(G)
        & m2_relset_1(G,k2_zfmisc_1(A,B),C) )
     => ! [H] :
          ( ( v1_funct_1(H)
            & m2_relset_1(H,k2_zfmisc_1(D,E),F) )
         => ( v1_funct_1(k3_funct_4(G,H))
            & m2_relset_1(k3_funct_4(G,H),k2_zfmisc_1(k2_zfmisc_1(A,D),k2_zfmisc_1(B,E)),k2_zfmisc_1(C,F)) ) ) ) )).

fof(t63_funct_4,axiom,(
    ! [A,B,C,D,E,F,G] :
      ( ( v1_funct_1(G)
        & v1_funct_2(G,k2_zfmisc_1(A,B),C)
        & m2_relset_1(G,k2_zfmisc_1(A,B),C) )
     => ! [H] :
          ( ( v1_funct_1(H)
            & v1_funct_2(H,k2_zfmisc_1(D,E),F)
            & m2_relset_1(H,k2_zfmisc_1(D,E),F) )
         => ~ ( C != k1_xboole_0
              & F != k1_xboole_0
              & ~ ( v1_funct_1(k3_funct_4(G,H))
                  & v1_funct_2(k3_funct_4(G,H),k2_zfmisc_1(k2_zfmisc_1(A,D),k2_zfmisc_1(B,E)),k2_zfmisc_1(C,F))
                  & m2_relset_1(k3_funct_4(G,H),k2_zfmisc_1(k2_zfmisc_1(A,D),k2_zfmisc_1(B,E)),k2_zfmisc_1(C,F)) ) ) ) ) )).

fof(t64_funct_4,axiom,(
    ! [A,B,C,D,E] :
      ( ~ v1_xboole_0(E)
     => ! [F] :
          ( ~ v1_xboole_0(F)
         => ! [G] :
              ( ( v1_funct_1(G)
                & v1_funct_2(G,k2_zfmisc_1(A,B),E)
                & m2_relset_1(G,k2_zfmisc_1(A,B),E) )
             => ! [H] :
                  ( ( v1_funct_1(H)
                    & v1_funct_2(H,k2_zfmisc_1(C,D),F)
                    & m2_relset_1(H,k2_zfmisc_1(C,D),F) )
                 => ( v1_funct_1(k3_funct_4(G,H))
                    & v1_funct_2(k3_funct_4(G,H),k2_zfmisc_1(k2_zfmisc_1(A,C),k2_zfmisc_1(B,D)),k2_zfmisc_1(E,F))
                    & m2_relset_1(k3_funct_4(G,H),k2_zfmisc_1(k2_zfmisc_1(A,C),k2_zfmisc_1(B,D)),k2_zfmisc_1(E,F)) ) ) ) ) ) )).

fof(d4_funct_4,axiom,(
    ! [A,B,C,D] : k4_funct_4(A,B,C,D) = k1_funct_4(k2_funcop_1(k1_tarski(A),C),k2_funcop_1(k1_tarski(B),D)) )).

fof(t65_funct_4,axiom,(
    ! [A,B,C,D] :
      ( k1_relat_1(k4_funct_4(A,B,C,D)) = k2_tarski(A,B)
      & r1_tarski(k2_relat_1(k4_funct_4(A,B,C,D)),k2_tarski(C,D)) ) )).

fof(t66_funct_4,axiom,(
    ! [A,B,C,D] :
      ( A != B
     => ( k1_funct_1(k4_funct_4(A,B,C,D),A) = C
        & k1_funct_1(k4_funct_4(A,B,C,D),B) = D ) ) )).

fof(t67_funct_4,axiom,(
    ! [A,B,C,D] :
      ( A != B
     => k2_relat_1(k4_funct_4(A,B,C,D)) = k2_tarski(C,D) ) )).

fof(t68_funct_4,axiom,(
    ! [A,B,C] : k4_funct_4(A,B,C,C) = k2_funcop_1(k2_tarski(A,B),C) )).

fof(t69_funct_4,axiom,(
    ! [A,B,C,D,E] :
      ( ( v1_relat_1(E)
        & v1_funct_1(E) )
     => ( ( k1_relat_1(E) = k2_tarski(A,B)
          & k1_funct_1(E,A) = C
          & k1_funct_1(E,B) = D )
       => E = k4_funct_4(A,B,C,D) ) ) )).

fof(t70_funct_4,axiom,(
    ! [A,B] : k2_funcop_1(k1_tarski(A),B) = k1_tarski(k4_tarski(A,B)) )).

fof(t71_funct_4,axiom,(
    ! [A,B,C,D] :
      ( A != C
     => k4_funct_4(A,C,B,D) = k2_tarski(k4_tarski(A,B),k4_tarski(C,D)) ) )).

fof(t72_funct_4,axiom,(
    ! [A,B,C,D,E,F] :
      ( k4_funct_4(A,B,C,D) = k4_funct_4(A,B,E,F)
     => ( A = B
        | ( C = E
          & D = F ) ) ) )).

fof(t73_funct_4,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) )
             => ! [D] :
                  ( ( v1_relat_1(D)
                    & v1_funct_1(D) )
                 => ( ( r1_tarski(k2_relat_1(C),k1_relat_1(A))
                      & r1_tarski(k2_relat_1(D),k1_relat_1(B))
                      & r1_partfun1(A,B) )
                   => k5_relat_1(k1_funct_4(C,D),k1_funct_4(A,B)) = k1_funct_4(k5_relat_1(C,A),k5_relat_1(D,B)) ) ) ) ) ) )).

fof(t74_funct_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B,C] :
          ( r1_tarski(k1_relat_1(A),k2_xboole_0(B,C))
         => k1_funct_4(k7_relat_1(A,B),k7_relat_1(A,C)) = A ) ) )).

fof(t75_funct_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ! [C] : k7_relat_1(k1_funct_4(A,B),C) = k1_funct_4(k7_relat_1(A,C),k7_relat_1(B,C)) ) ) )).

fof(t76_funct_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ! [C] :
              ( r1_xboole_0(C,k1_relat_1(B))
             => k7_relat_1(k1_funct_4(A,B),C) = k7_relat_1(A,C) ) ) ) )).

fof(t77_funct_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ! [C] :
              ( r1_xboole_0(k1_relat_1(A),C)
             => k7_relat_1(k1_funct_4(A,B),C) = k7_relat_1(B,C) ) ) ) )).

fof(t78_funct_4,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(B) = k1_relat_1(C)
               => k1_funct_4(k1_funct_4(A,B),C) = k1_funct_4(A,C) ) ) ) ) )).

fof(t79_funct_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( r1_tarski(A,B)
           => ( k1_funct_4(A,B) = B
              & k1_funct_4(B,A) = B ) ) ) ) )).

fof(t80_funct_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : k1_funct_4(A,k7_relat_1(A,B)) = A ) )).

fof(t81_funct_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ! [C,D] :
              ( ( r1_tarski(k1_relat_1(A),C)
                & r1_tarski(k1_relat_1(B),D)
                & r1_xboole_0(C,D) )
             => ( k7_relat_1(k1_funct_4(A,B),C) = A
                & k7_relat_1(k1_funct_4(A,B),D) = B ) ) ) ) )).

fof(t82_funct_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ! [C] :
              ( ( r1_tarski(k1_relat_1(A),C)
                & r1_xboole_0(k1_relat_1(B),C) )
             => k7_relat_1(k1_funct_4(A,B),C) = A ) ) ) )).

fof(t83_funct_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B,C] : k7_relat_1(A,k2_xboole_0(B,C)) = k1_funct_4(k7_relat_1(A,B),k7_relat_1(A,C)) ) )).

fof(dt_k1_funct_4,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_relat_1(B)
        & v1_funct_1(B) )
     => ( v1_relat_1(k1_funct_4(A,B))
        & v1_funct_1(k1_funct_4(A,B)) ) ) )).

fof(idempotence_k1_funct_4,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_relat_1(B)
        & v1_funct_1(B) )
     => k1_funct_4(A,A) = A ) )).

fof(dt_k2_funct_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v1_relat_1(k2_funct_4(A))
        & v1_funct_1(k2_funct_4(A)) ) ) )).

fof(dt_k3_funct_4,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_relat_1(B)
        & v1_funct_1(B) )
     => ( v1_relat_1(k3_funct_4(A,B))
        & v1_funct_1(k3_funct_4(A,B)) ) ) )).

fof(dt_k4_funct_4,axiom,(
    $true )).

fof(dt_k5_funct_4,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(D,A)
        & m1_subset_1(E,A) )
     => ( v1_funct_1(k5_funct_4(A,B,C,D,E))
        & v1_funct_2(k5_funct_4(A,B,C,D,E),k2_tarski(B,C),A)
        & m2_relset_1(k5_funct_4(A,B,C,D,E),k2_tarski(B,C),A) ) ) )).

fof(redefinition_k5_funct_4,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(D,A)
        & m1_subset_1(E,A) )
     => k5_funct_4(A,B,C,D,E) = k4_funct_4(B,C,D,E) ) )).
%------------------------------------------------------------------------------