## SET007 Axioms: SET007+21.ax

```%------------------------------------------------------------------------------
% File     : SET007+21 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Basic Functions and Operations on 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_3 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  135 (  32 unit)
%            Number of atoms       :  648 ( 127 equality)
%            Maximal formula depth :   17 (   6 average)
%            Number of connectives :  548 (  35 ~  ;   2  |; 288  &)
%                                         (   8 <=>; 215 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   13 (   1 propositional; 0-3 arity)
%            Number of functors    :   44 (   7 constant; 0-6 arity)
%            Number of variables   :  342 (   0 singleton; 340 !;   2 ?)
%            Maximal term depth    :    5 (   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(t1_funct_3,axiom,(
! [A,B] :
( r1_tarski(A,B)
=> k6_relat_1(A) = k7_relat_1(k6_relat_1(B),A) ) )).

fof(t2_funct_3,axiom,(
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( r1_tarski(A,k1_relat_1(k5_relat_1(B,C)))
=> r1_tarski(k9_relat_1(B,A),k1_relat_1(C)) ) ) ) )).

fof(t3_funct_3,axiom,(
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( ( r1_tarski(A,k1_relat_1(B))
& r1_tarski(k9_relat_1(B,A),k1_relat_1(C)) )
=> r1_tarski(A,k1_relat_1(k5_relat_1(B,C))) ) ) ) )).

fof(t4_funct_3,axiom,(
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( ( r1_tarski(A,k2_relat_1(k5_relat_1(B,C)))
& v2_funct_1(C) )
=> r1_tarski(k10_relat_1(C,A),k2_relat_1(B)) ) ) ) )).

fof(t5_funct_3,axiom,(
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( ( r1_tarski(A,k2_relat_1(C))
& r1_tarski(k10_relat_1(C,A),k2_relat_1(B)) )
=> r1_tarski(A,k2_relat_1(k5_relat_1(B,C))) ) ) ) )).

fof(t6_funct_3,axiom,(
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ( ( k1_relat_1(C) = k2_zfmisc_1(A,B)
& k1_relat_1(D) = k2_zfmisc_1(A,B)
& ! [E,F] :
( ( r2_hidden(E,A)
& r2_hidden(F,B) )
=> k1_funct_1(C,k4_tarski(E,F)) = k1_funct_1(D,k4_tarski(E,F)) ) )
=> C = D ) ) ) )).

fof(d1_funct_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( B = k1_funct_3(A)
<=> ( k1_relat_1(B) = k1_zfmisc_1(k1_relat_1(A))
& ! [C] :
( r1_tarski(C,k1_relat_1(A))
=> k1_funct_1(B,C) = k9_relat_1(A,C) ) ) ) ) ) )).

fof(t7_funct_3,axiom,(
\$true )).

fof(t8_funct_3,axiom,(
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r2_hidden(A,k1_relat_1(k1_funct_3(B)))
=> k1_funct_1(k1_funct_3(B),A) = k9_relat_1(B,A) ) ) )).

fof(t9_funct_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> k1_funct_1(k1_funct_3(A),k1_xboole_0) = k1_xboole_0 ) )).

fof(t10_funct_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> r1_tarski(k2_relat_1(k1_funct_3(A)),k1_zfmisc_1(k2_relat_1(A))) ) )).

fof(t11_funct_3,axiom,(
\$true )).

fof(t12_funct_3,axiom,(
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> r1_tarski(k9_relat_1(k1_funct_3(B),A),k1_zfmisc_1(k2_relat_1(B))) ) )).

fof(t13_funct_3,axiom,(
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> r1_tarski(k10_relat_1(k1_funct_3(B),A),k1_zfmisc_1(k1_relat_1(B))) ) )).

fof(t14_funct_3,axiom,(
! [A,B,C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,C)
& m2_relset_1(D,A,C) )
=> r1_tarski(k10_relat_1(k1_funct_3(D),B),k1_zfmisc_1(A)) ) ) )).

fof(t15_funct_3,axiom,(
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> r1_tarski(k3_tarski(k9_relat_1(k1_funct_3(B),A)),k9_relat_1(B,k3_tarski(A))) ) )).

fof(t16_funct_3,axiom,(
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r1_tarski(A,k1_zfmisc_1(k1_relat_1(B)))
=> k9_relat_1(B,k3_tarski(A)) = k3_tarski(k9_relat_1(k1_funct_3(B),A)) ) ) )).

fof(t17_funct_3,axiom,(
! [A,B,C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,C)
& m2_relset_1(D,A,C) )
=> ( r1_tarski(B,k1_zfmisc_1(A))
=> k2_funct_2(A,C,D,k3_tarski(B)) = k3_tarski(k9_relat_1(k1_funct_3(D),B)) ) ) ) )).

fof(t18_funct_3,axiom,(
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> r1_tarski(k3_tarski(k10_relat_1(k1_funct_3(B),A)),k10_relat_1(B,k3_tarski(A))) ) )).

fof(t19_funct_3,axiom,(
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r1_tarski(A,k1_zfmisc_1(k2_relat_1(B)))
=> k10_relat_1(B,k3_tarski(A)) = k3_tarski(k10_relat_1(k1_funct_3(B),A)) ) ) )).

fof(t20_funct_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> k1_funct_3(k5_relat_1(A,B)) = k5_relat_1(k1_funct_3(A),k1_funct_3(B)) ) ) )).

fof(t21_funct_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v1_funct_1(k1_funct_3(A))
& v1_funct_2(k1_funct_3(A),k1_zfmisc_1(k1_relat_1(A)),k1_zfmisc_1(k2_relat_1(A)))
& m2_relset_1(k1_funct_3(A),k1_zfmisc_1(k1_relat_1(A)),k1_zfmisc_1(k2_relat_1(A))) ) ) )).

fof(t22_funct_3,axiom,(
! [A,B,C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,B)
& m2_relset_1(C,A,B) )
=> ( ( B = k1_xboole_0
=> A = k1_xboole_0 )
=> ( v1_funct_1(k1_funct_3(C))
& v1_funct_2(k1_funct_3(C),k1_zfmisc_1(A),k1_zfmisc_1(B))
& m2_relset_1(k1_funct_3(C),k1_zfmisc_1(A),k1_zfmisc_1(B)) ) ) ) )).

fof(d2_funct_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( B = k3_funct_3(A)
<=> ( k1_relat_1(B) = k1_zfmisc_1(k2_relat_1(A))
& ! [C] :
( r1_tarski(C,k2_relat_1(A))
=> k1_funct_1(B,C) = k10_relat_1(A,C) ) ) ) ) ) )).

fof(t23_funct_3,axiom,(
\$true )).

fof(t24_funct_3,axiom,(
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r2_hidden(A,k1_relat_1(k3_funct_3(B)))
=> k1_funct_1(k3_funct_3(B),A) = k10_relat_1(B,A) ) ) )).

fof(t25_funct_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> r1_tarski(k2_relat_1(k3_funct_3(A)),k1_zfmisc_1(k1_relat_1(A))) ) )).

fof(t26_funct_3,axiom,(
\$true )).

fof(t27_funct_3,axiom,(
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> r1_tarski(k9_relat_1(k3_funct_3(B),A),k1_zfmisc_1(k1_relat_1(B))) ) )).

fof(t28_funct_3,axiom,(
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> r1_tarski(k10_relat_1(k3_funct_3(B),A),k1_zfmisc_1(k2_relat_1(B))) ) )).

fof(t29_funct_3,axiom,(
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> r1_tarski(k3_tarski(k9_relat_1(k3_funct_3(B),A)),k10_relat_1(B,k3_tarski(A))) ) )).

fof(t30_funct_3,axiom,(
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r1_tarski(A,k1_zfmisc_1(k2_relat_1(B)))
=> k3_tarski(k9_relat_1(k3_funct_3(B),A)) = k10_relat_1(B,k3_tarski(A)) ) ) )).

fof(t31_funct_3,axiom,(
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> r1_tarski(k3_tarski(k10_relat_1(k3_funct_3(B),A)),k9_relat_1(B,k3_tarski(A))) ) )).

fof(t32_funct_3,axiom,(
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( ( r1_tarski(A,k1_zfmisc_1(k1_relat_1(B)))
& v2_funct_1(B) )
=> k3_tarski(k10_relat_1(k3_funct_3(B),A)) = k9_relat_1(B,k3_tarski(A)) ) ) )).

fof(t33_funct_3,axiom,(
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> r1_tarski(k9_relat_1(k3_funct_3(B),A),k10_relat_1(k1_funct_3(B),A)) ) )).

fof(t34_funct_3,axiom,(
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( v2_funct_1(B)
=> k9_relat_1(k3_funct_3(B),A) = k10_relat_1(k1_funct_3(B),A) ) ) )).

fof(t35_funct_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( r1_tarski(B,k1_zfmisc_1(k1_relat_1(A)))
=> r1_tarski(k10_relat_1(k3_funct_3(A),B),k9_relat_1(k1_funct_3(A),B)) ) ) )).

fof(t36_funct_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( v2_funct_1(A)
=> r1_tarski(k9_relat_1(k1_funct_3(A),B),k10_relat_1(k3_funct_3(A),B)) ) ) )).

fof(t37_funct_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v2_funct_1(A)
& r1_tarski(B,k1_zfmisc_1(k1_relat_1(A))) )
=> k10_relat_1(k3_funct_3(A),B) = k9_relat_1(k1_funct_3(A),B) ) ) )).

fof(t38_funct_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( v2_funct_1(B)
=> k3_funct_3(k5_relat_1(A,B)) = k5_relat_1(k3_funct_3(B),k3_funct_3(A)) ) ) ) )).

fof(t39_funct_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v1_funct_1(k3_funct_3(A))
& v1_funct_2(k3_funct_3(A),k1_zfmisc_1(k2_relat_1(A)),k1_zfmisc_1(k1_relat_1(A)))
& m2_relset_1(k3_funct_3(A),k1_zfmisc_1(k2_relat_1(A)),k1_zfmisc_1(k1_relat_1(A))) ) ) )).

fof(d3_funct_3,axiom,(
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( C = k4_funct_3(A,B)
<=> ( k1_relat_1(C) = B
& ! [D] :
( r2_hidden(D,B)
=> ( ( r2_hidden(D,A)
=> k1_funct_1(C,D) = np__1 )
& ( ~ r2_hidden(D,A)
=> k1_funct_1(C,D) = np__0 ) ) ) ) ) ) )).

fof(t40_funct_3,axiom,(
\$true )).

fof(t41_funct_3,axiom,(
\$true )).

fof(t42_funct_3,axiom,(
! [A,B,C] :
( k1_funct_1(k4_funct_3(B,C),A) = np__1
=> r2_hidden(A,B) ) )).

fof(t43_funct_3,axiom,(
! [A,B,C] :
( r2_hidden(A,k4_xboole_0(B,C))
=> k1_funct_1(k4_funct_3(C,B),A) = np__0 ) )).

fof(t44_funct_3,axiom,(
\$true )).

fof(t45_funct_3,axiom,(
\$true )).

fof(t46_funct_3,axiom,(
\$true )).

fof(t47_funct_3,axiom,(
! [A,B,C] :
( ( r1_tarski(A,B)
& r1_tarski(C,B)
& k4_funct_3(A,B) = k4_funct_3(C,B) )
=> A = C ) )).

fof(t48_funct_3,axiom,(
! [A,B] : r1_tarski(k2_relat_1(k4_funct_3(A,B)),k2_tarski(np__0,np__1)) )).

fof(t49_funct_3,axiom,(
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k2_tarski(np__0,np__1))
& m2_relset_1(B,A,k2_tarski(np__0,np__1)) )
=> B = k4_funct_3(k3_funct_2(A,k2_tarski(np__0,np__1),B,k1_tarski(np__1)),A) ) )).

fof(t50_funct_3,axiom,(
\$true )).

fof(t51_funct_3,axiom,(
\$true )).

fof(t52_funct_3,axiom,(
\$true )).

fof(t53_funct_3,axiom,(
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> k6_funct_3(A,B) = k7_relat_1(k6_relat_1(A),B) ) )).

fof(t54_funct_3,axiom,(
\$true )).

fof(t55_funct_3,axiom,(
\$true )).

fof(t56_funct_3,axiom,(
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(B))
=> ( r2_hidden(A,C)
=> r2_hidden(k1_funct_1(k6_funct_3(B,C),A),B) ) ) )).

fof(d4_funct_3,axiom,(
\$true )).

fof(d5_funct_3,axiom,(
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( C = k7_funct_3(A,B)
<=> ( k1_relat_1(C) = k2_zfmisc_1(A,B)
& ! [D,E] :
( ( r2_hidden(D,A)
& r2_hidden(E,B) )
=> k1_funct_1(C,k4_tarski(D,E)) = D ) ) ) ) )).

fof(d6_funct_3,axiom,(
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( C = k8_funct_3(A,B)
<=> ( k1_relat_1(C) = k2_zfmisc_1(A,B)
& ! [D,E] :
( ( r2_hidden(D,A)
& r2_hidden(E,B) )
=> k1_funct_1(C,k4_tarski(D,E)) = E ) ) ) ) )).

fof(t57_funct_3,axiom,(
\$true )).

fof(t58_funct_3,axiom,(
\$true )).

fof(t59_funct_3,axiom,(
! [A,B] : r1_tarski(k2_relat_1(k7_funct_3(A,B)),A) )).

fof(t60_funct_3,axiom,(
! [A,B] :
( A != k1_xboole_0
=> k2_relat_1(k7_funct_3(B,A)) = B ) )).

fof(t61_funct_3,axiom,(
! [A,B] : r1_tarski(k2_relat_1(k8_funct_3(A,B)),B) )).

fof(t62_funct_3,axiom,(
! [A,B] :
( A != k1_xboole_0
=> k2_relat_1(k8_funct_3(A,B)) = B ) )).

fof(d7_funct_3,axiom,(
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( B = k11_funct_3(A)
<=> ( k1_relat_1(B) = A
& ! [C] :
( r2_hidden(C,A)
=> k1_funct_1(B,C) = k4_tarski(C,C) ) ) ) ) )).

fof(t63_funct_3,axiom,(
\$true )).

fof(t64_funct_3,axiom,(
\$true )).

fof(t65_funct_3,axiom,(
\$true )).

fof(t66_funct_3,axiom,(
! [A] : r1_tarski(k2_relat_1(k11_funct_3(A)),k2_zfmisc_1(A,A)) )).

fof(d8_funct_3,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 = k13_funct_3(A,B)
<=> ( k1_relat_1(C) = k3_xboole_0(k1_relat_1(A),k1_relat_1(B))
& ! [D] :
( r2_hidden(D,k1_relat_1(C))
=> k1_funct_1(C,D) = k4_tarski(k1_funct_1(A,D),k1_funct_1(B,D)) ) ) ) ) ) ) )).

fof(t67_funct_3,axiom,(
\$true )).

fof(t68_funct_3,axiom,(
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( r2_hidden(A,k3_xboole_0(k1_relat_1(B),k1_relat_1(C)))
=> k1_funct_1(k13_funct_3(B,C),A) = k4_tarski(k1_funct_1(B,A),k1_funct_1(C,A)) ) ) ) )).

fof(t69_funct_3,axiom,(
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ( ( k1_relat_1(C) = B
& k1_relat_1(D) = B
& r2_hidden(A,B) )
=> k1_funct_1(k13_funct_3(C,D),A) = k4_tarski(k1_funct_1(C,A),k1_funct_1(D,A)) ) ) ) )).

fof(t70_funct_3,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 )
=> k1_relat_1(k13_funct_3(B,C)) = A ) ) ) )).

fof(t71_funct_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> r1_tarski(k2_relat_1(k13_funct_3(A,B)),k2_zfmisc_1(k2_relat_1(A),k2_relat_1(B))) ) ) )).

fof(t72_funct_3,axiom,(
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ( ( k1_relat_1(C) = k1_relat_1(D)
& r1_tarski(k2_relat_1(C),A)
& r1_tarski(k2_relat_1(D),B) )
=> ( k5_relat_1(k13_funct_3(C,D),k9_funct_3(A,B)) = C
& k5_relat_1(k13_funct_3(C,D),k10_funct_3(A,B)) = D ) ) ) ) )).

fof(t73_funct_3,axiom,(
! [A,B] : k13_funct_3(k9_funct_3(A,B),k10_funct_3(A,B)) = k6_relat_1(k2_zfmisc_1(A,B)) )).

fof(t74_funct_3,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) )
=> ( ( k1_relat_1(A) = k1_relat_1(B)
& k1_relat_1(D) = k1_relat_1(C)
& k13_funct_3(A,B) = k13_funct_3(D,C) )
=> ( A = D
& B = C ) ) ) ) ) ) )).

fof(t75_funct_3,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) )
=> k13_funct_3(k5_relat_1(C,A),k5_relat_1(C,B)) = k5_relat_1(C,k13_funct_3(A,B)) ) ) ) )).

fof(t76_funct_3,axiom,(
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> r1_tarski(k9_relat_1(k13_funct_3(B,C),A),k2_zfmisc_1(k9_relat_1(B,A),k9_relat_1(C,A))) ) ) )).

fof(t77_funct_3,axiom,(
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> k10_relat_1(k13_funct_3(C,D),k2_zfmisc_1(A,B)) = k3_xboole_0(k10_relat_1(C,A),k10_relat_1(D,B)) ) ) ) )).

fof(t78_funct_3,axiom,(
! [A,B,C,D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,B)
& m2_relset_1(D,A,B) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,A,C)
& m2_relset_1(E,A,C) )
=> ~ ( ( B = k1_xboole_0
=> A = k1_xboole_0 )
& ( C = k1_xboole_0
=> A = k1_xboole_0 )
& ~ ( v1_funct_1(k13_funct_3(D,E))
& v1_funct_2(k13_funct_3(D,E),A,k2_zfmisc_1(B,C))
& m2_relset_1(k13_funct_3(D,E),A,k2_zfmisc_1(B,C)) ) ) ) ) )).

fof(t79_funct_3,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,B)
& m2_relset_1(D,A,B) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,A,C)
& m2_relset_1(E,A,C) )
=> ! [F] :
( m1_subset_1(F,A)
=> k8_funct_2(A,k2_zfmisc_1(B,C),k14_funct_3(A,B,C,D,E),F) = k4_tarski(k8_funct_2(A,B,D,F),k8_funct_2(A,C,E,F)) ) ) ) ) ) ) )).

fof(t80_funct_3,axiom,(
! [A,B,C,D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,B)
& m2_relset_1(D,A,B) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,A,C)
& m2_relset_1(E,A,C) )
=> r1_tarski(k2_relat_1(k13_funct_3(D,E)),k2_zfmisc_1(B,C)) ) ) )).

fof(t81_funct_3,axiom,(
! [A,B,C,D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,B)
& m2_relset_1(D,A,B) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,A,C)
& m2_relset_1(E,A,C) )
=> ~ ( ( B = k1_xboole_0
=> A = k1_xboole_0 )
& ( C = k1_xboole_0
=> A = k1_xboole_0 )
& ~ ( k5_relat_1(k13_funct_3(D,E),k9_funct_3(B,C)) = D
& k5_relat_1(k13_funct_3(D,E),k10_funct_3(B,C)) = E ) ) ) ) )).

fof(t82_funct_3,axiom,(
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,B)
& m2_relset_1(D,A,B) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,A,C)
& m2_relset_1(E,A,C) )
=> ( k7_funct_2(A,k2_zfmisc_1(B,C),B,k14_funct_3(A,B,C,D,E),k9_funct_3(B,C)) = D
& k7_funct_2(A,k2_zfmisc_1(B,C),C,k14_funct_3(A,B,C,D,E),k10_funct_3(B,C)) = E ) ) ) ) ) )).

fof(t83_funct_3,axiom,(
! [A,B,C,D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,B)
& m2_relset_1(D,A,B) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,A,B)
& m2_relset_1(E,A,B) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,A,C)
& m2_relset_1(F,A,C) )
=> ! [G] :
( ( v1_funct_1(G)
& v1_funct_2(G,A,C)
& m2_relset_1(G,A,C) )
=> ( k13_funct_3(D,F) = k13_funct_3(E,G)
=> ( ( B = k1_xboole_0
& A != k1_xboole_0 )
| ( C = k1_xboole_0
& A != k1_xboole_0 )
| ( D = E
& F = G ) ) ) ) ) ) ) )).

fof(t84_funct_3,axiom,(
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,B)
& m2_relset_1(D,A,B) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,A,B)
& m2_relset_1(E,A,B) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,A,C)
& m2_relset_1(F,A,C) )
=> ! [G] :
( ( v1_funct_1(G)
& v1_funct_2(G,A,C)
& m2_relset_1(G,A,C) )
=> ( k14_funct_3(A,B,C,D,F) = k14_funct_3(A,B,C,E,G)
=> ( D = E
& F = G ) ) ) ) ) ) ) ) )).

fof(d9_funct_3,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 = k15_funct_3(A,B)
<=> ( k1_relat_1(C) = k2_zfmisc_1(k1_relat_1(A),k1_relat_1(B))
& ! [D,E] :
( ( r2_hidden(D,k1_relat_1(A))
& r2_hidden(E,k1_relat_1(B)) )
=> k1_funct_1(C,k4_tarski(D,E)) = k4_tarski(k1_funct_1(A,D),k1_funct_1(B,E)) ) ) ) ) ) ) )).

fof(t85_funct_3,axiom,(
\$true )).

fof(t86_funct_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C,D] :
( r2_hidden(k4_tarski(C,D),k2_zfmisc_1(k1_relat_1(A),k1_relat_1(B)))
=> k1_funct_1(k15_funct_3(A,B),k4_tarski(C,D)) = k4_tarski(k1_funct_1(A,C),k1_funct_1(B,D)) ) ) ) )).

fof(t87_funct_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> k15_funct_3(A,B) = k13_funct_3(k5_relat_1(k9_funct_3(k1_relat_1(A),k1_relat_1(B)),A),k5_relat_1(k10_funct_3(k1_relat_1(A),k1_relat_1(B)),B)) ) ) )).

fof(t88_funct_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> k2_relat_1(k15_funct_3(A,B)) = k2_zfmisc_1(k2_relat_1(A),k2_relat_1(B)) ) ) )).

fof(t89_funct_3,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 )
=> k13_funct_3(B,C) = k5_relat_1(k12_funct_3(A),k15_funct_3(B,C)) ) ) ) )).

fof(t90_funct_3,axiom,(
! [A,B] : k15_funct_3(k6_relat_1(A),k6_relat_1(B)) = k6_relat_1(k2_zfmisc_1(A,B)) )).

fof(t91_funct_3,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) )
=> k5_relat_1(k13_funct_3(B,D),k15_funct_3(A,C)) = k13_funct_3(k5_relat_1(B,A),k5_relat_1(D,C)) ) ) ) ) )).

fof(t92_funct_3,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) )
=> k5_relat_1(k15_funct_3(B,D),k15_funct_3(A,C)) = k15_funct_3(k5_relat_1(B,A),k5_relat_1(D,C)) ) ) ) ) )).

fof(t93_funct_3,axiom,(
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> k9_relat_1(k15_funct_3(C,D),k2_zfmisc_1(A,B)) = k2_zfmisc_1(k9_relat_1(C,A),k9_relat_1(D,B)) ) ) )).

fof(t94_funct_3,axiom,(
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> k10_relat_1(k15_funct_3(C,D),k2_zfmisc_1(A,B)) = k2_zfmisc_1(k10_relat_1(C,A),k10_relat_1(D,B)) ) ) )).

fof(t95_funct_3,axiom,(
! [A,B,C,D,E] :
( ( v1_funct_1(E)
& v1_funct_2(E,A,B)
& m2_relset_1(E,A,B) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,C,D)
& m2_relset_1(F,C,D) )
=> ( v1_funct_1(k15_funct_3(E,F))
& v1_funct_2(k15_funct_3(E,F),k2_zfmisc_1(A,C),k2_zfmisc_1(B,D))
& m2_relset_1(k15_funct_3(E,F),k2_zfmisc_1(A,C),k2_zfmisc_1(B,D)) ) ) ) )).

fof(t96_funct_3,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ~ v1_xboole_0(D)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,A,B)
& m2_relset_1(E,A,B) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,C,D)
& m2_relset_1(F,C,D) )
=> ! [G] :
( m1_subset_1(G,A)
=> ! [H] :
( m1_subset_1(H,C)
=> k1_funct_1(k16_funct_3(A,C,B,D,E,F),k4_tarski(G,H)) = k4_tarski(k8_funct_2(A,B,E,G),k8_funct_2(C,D,F,H)) ) ) ) ) ) ) ) ) )).

fof(t97_funct_3,axiom,(
! [A,B,C,D,E] :
( ( v1_funct_1(E)
& v1_funct_2(E,A,B)
& m2_relset_1(E,A,B) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,C,D)
& m2_relset_1(F,C,D) )
=> ~ ( ( B = k1_xboole_0
=> A = k1_xboole_0 )
& ( D = k1_xboole_0
=> C = k1_xboole_0 )
& k16_funct_3(A,C,B,D,E,F) != k13_funct_3(k5_relat_1(k9_funct_3(A,C),E),k5_relat_1(k10_funct_3(A,C),F)) ) ) ) )).

fof(t98_funct_3,axiom,(
! [A,B,C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ~ v1_xboole_0(D)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,A,C)
& m2_relset_1(E,A,C) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,B,D)
& m2_relset_1(F,B,D) )
=> k16_funct_3(A,B,C,D,E,F) = k13_funct_3(k5_relat_1(k9_funct_3(A,B),E),k5_relat_1(k10_funct_3(A,B),F)) ) ) ) ) )).

fof(t99_funct_3,axiom,(
! [A,B,C,D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,B)
& m2_relset_1(D,A,B) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,A,C)
& m2_relset_1(E,A,C) )
=> k13_funct_3(D,E) = k5_relat_1(k12_funct_3(A),k16_funct_3(A,A,B,C,D,E)) ) ) )).

fof(t100_funct_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> k2_funct_2(k2_zfmisc_1(k1_relat_1(A),k2_relat_1(A)),k1_relat_1(A),k9_funct_3(k1_relat_1(A),k2_relat_1(A)),A) = k1_relat_1(A) ) )).

fof(s1_funct_3,axiom,
( ( ! [A,B,C,D] :
( ( r2_hidden(A,f1_s1_funct_3)
& r2_hidden(B,f2_s1_funct_3)
& p1_s1_funct_3(A,B,C)
& p1_s1_funct_3(A,B,D) )
=> C = D )
& ! [A,B] :
~ ( r2_hidden(A,f1_s1_funct_3)
& r2_hidden(B,f2_s1_funct_3)
& ! [C] : ~ p1_s1_funct_3(A,B,C) ) )
=> ? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& k1_relat_1(A) = k2_zfmisc_1(f1_s1_funct_3,f2_s1_funct_3)
& ! [B,C] :
( ( r2_hidden(B,f1_s1_funct_3)
& r2_hidden(C,f2_s1_funct_3) )
=> p1_s1_funct_3(B,C,k1_funct_1(A,k4_tarski(B,C))) ) ) )).

fof(s2_funct_3,axiom,(
? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& k1_relat_1(A) = k2_zfmisc_1(f1_s2_funct_3,f2_s2_funct_3)
& ! [B,C] :
( ( r2_hidden(B,f1_s2_funct_3)
& r2_hidden(C,f2_s2_funct_3) )
=> k1_funct_1(A,k4_tarski(B,C)) = f3_s2_funct_3(B,C) ) ) )).

fof(dt_k1_funct_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v1_relat_1(k1_funct_3(A))
& v1_funct_1(k1_funct_3(A)) ) ) )).

fof(dt_k2_funct_3,axiom,(
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& v1_funct_1(C)
& v1_funct_2(C,A,B)
& m1_relset_1(C,A,B) )
=> ( v1_funct_1(k2_funct_3(A,B,C))
& v1_funct_2(k2_funct_3(A,B,C),k1_zfmisc_1(A),k1_zfmisc_1(B))
& m2_relset_1(k2_funct_3(A,B,C),k1_zfmisc_1(A),k1_zfmisc_1(B)) ) ) )).

fof(redefinition_k2_funct_3,axiom,(
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& v1_funct_1(C)
& v1_funct_2(C,A,B)
& m1_relset_1(C,A,B) )
=> k2_funct_3(A,B,C) = k1_funct_3(C) ) )).

fof(dt_k3_funct_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v1_relat_1(k3_funct_3(A))
& v1_funct_1(k3_funct_3(A)) ) ) )).

fof(dt_k4_funct_3,axiom,(
! [A,B] :
( v1_relat_1(k4_funct_3(A,B))
& v1_funct_1(k4_funct_3(A,B)) ) )).

fof(dt_k5_funct_3,axiom,(
! [A,B] :
( v1_funct_1(k5_funct_3(A,B))
& v1_funct_2(k5_funct_3(A,B),B,k2_tarski(np__0,np__1))
& m2_relset_1(k5_funct_3(A,B),B,k2_tarski(np__0,np__1)) ) )).

fof(redefinition_k5_funct_3,axiom,(
! [A,B] : k5_funct_3(A,B) = k4_funct_3(A,B) )).

fof(dt_k6_funct_3,axiom,(
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ( v1_funct_1(k6_funct_3(A,B))
& v1_funct_2(k6_funct_3(A,B),B,A)
& m2_relset_1(k6_funct_3(A,B),B,A) ) ) )).

fof(redefinition_k6_funct_3,axiom,(
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> k6_funct_3(A,B) = k6_relat_1(B) ) )).

fof(dt_k7_funct_3,axiom,(
! [A,B] :
( v1_relat_1(k7_funct_3(A,B))
& v1_funct_1(k7_funct_3(A,B)) ) )).

fof(dt_k8_funct_3,axiom,(
! [A,B] :
( v1_relat_1(k8_funct_3(A,B))
& v1_funct_1(k8_funct_3(A,B)) ) )).

fof(dt_k9_funct_3,axiom,(
! [A,B] :
( v1_funct_1(k9_funct_3(A,B))
& v1_funct_2(k9_funct_3(A,B),k2_zfmisc_1(A,B),A)
& m2_relset_1(k9_funct_3(A,B),k2_zfmisc_1(A,B),A) ) )).

fof(redefinition_k9_funct_3,axiom,(
! [A,B] : k9_funct_3(A,B) = k7_funct_3(A,B) )).

fof(dt_k10_funct_3,axiom,(
! [A,B] :
( v1_funct_1(k10_funct_3(A,B))
& v1_funct_2(k10_funct_3(A,B),k2_zfmisc_1(A,B),B)
& m2_relset_1(k10_funct_3(A,B),k2_zfmisc_1(A,B),B) ) )).

fof(redefinition_k10_funct_3,axiom,(
! [A,B] : k10_funct_3(A,B) = k8_funct_3(A,B) )).

fof(dt_k11_funct_3,axiom,(
! [A] :
( v1_relat_1(k11_funct_3(A))
& v1_funct_1(k11_funct_3(A)) ) )).

fof(dt_k12_funct_3,axiom,(
! [A] :
( v1_funct_1(k12_funct_3(A))
& v1_funct_2(k12_funct_3(A),A,k2_zfmisc_1(A,A))
& m2_relset_1(k12_funct_3(A),A,k2_zfmisc_1(A,A)) ) )).

fof(redefinition_k12_funct_3,axiom,(
! [A] : k12_funct_3(A) = k11_funct_3(A) )).

fof(dt_k13_funct_3,axiom,(
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_relat_1(B)
& v1_funct_1(B) )
=> ( v1_relat_1(k13_funct_3(A,B))
& v1_funct_1(k13_funct_3(A,B)) ) ) )).

fof(dt_k14_funct_3,axiom,(
! [A,B,C,D,E] :
( ( ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& v1_funct_1(D)
& v1_funct_2(D,A,B)
& m1_relset_1(D,A,B)
& v1_funct_1(E)
& v1_funct_2(E,A,C)
& m1_relset_1(E,A,C) )
=> ( v1_funct_1(k14_funct_3(A,B,C,D,E))
& v1_funct_2(k14_funct_3(A,B,C,D,E),A,k2_zfmisc_1(B,C))
& m2_relset_1(k14_funct_3(A,B,C,D,E),A,k2_zfmisc_1(B,C)) ) ) )).

fof(redefinition_k14_funct_3,axiom,(
! [A,B,C,D,E] :
( ( ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& v1_funct_1(D)
& v1_funct_2(D,A,B)
& m1_relset_1(D,A,B)
& v1_funct_1(E)
& v1_funct_2(E,A,C)
& m1_relset_1(E,A,C) )
=> k14_funct_3(A,B,C,D,E) = k13_funct_3(D,E) ) )).

fof(dt_k15_funct_3,axiom,(
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_relat_1(B)
& v1_funct_1(B) )
=> ( v1_relat_1(k15_funct_3(A,B))
& v1_funct_1(k15_funct_3(A,B)) ) ) )).

fof(dt_k16_funct_3,axiom,(
! [A,B,C,D,E,F] :
( ( v1_funct_1(E)
& v1_funct_2(E,A,C)
& m1_relset_1(E,A,C)
& v1_funct_1(F)
& v1_funct_2(F,B,D)
& m1_relset_1(F,B,D) )
=> ( v1_funct_1(k16_funct_3(A,B,C,D,E,F))
& v1_funct_2(k16_funct_3(A,B,C,D,E,F),k2_zfmisc_1(A,B),k2_zfmisc_1(C,D))
& m2_relset_1(k16_funct_3(A,B,C,D,E,F),k2_zfmisc_1(A,B),k2_zfmisc_1(C,D)) ) ) )).

fof(redefinition_k16_funct_3,axiom,(
! [A,B,C,D,E,F] :
( ( v1_funct_1(E)
& v1_funct_2(E,A,C)
& m1_relset_1(E,A,C)
& v1_funct_1(F)
& v1_funct_2(F,B,D)
& m1_relset_1(F,B,D) )
=> k16_funct_3(A,B,C,D,E,F) = k15_funct_3(E,F) ) )).
%------------------------------------------------------------------------------
```