SET007 Axioms: SET007+412.ax
%------------------------------------------------------------------------------
% File : SET007+412 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Categorial Categories and Slice Categories
% 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 : cat_5 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 99 ( 0 unt; 0 def)
% Number of atoms : 807 ( 135 equ)
% Maximal formula atoms : 62 ( 8 avg)
% Number of connectives : 744 ( 36 ~; 0 |; 359 &)
% ( 17 <=>; 332 =>; 0 <=; 0 <~>)
% Maximal formula depth : 30 ( 9 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 19 ( 18 usr; 0 prp; 1-3 aty)
% Number of functors : 68 ( 68 usr; 6 con; 0-6 aty)
% Number of variables : 379 ( 351 !; 28 ?)
% 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(rc1_cat_5,axiom,
? [A] :
( l1_cat_1(A)
& v1_cat_1(A)
& v2_cat_1(A)
& v1_cat_5(A) ) ).
fof(rc2_cat_5,axiom,
? [A] :
( ~ v1_xboole_0(A)
& v2_cat_5(A) ) ).
fof(cc1_cat_5,axiom,
! [A] :
( l1_cat_1(A)
=> ( ( v2_cat_1(A)
& v3_cat_5(A) )
=> ( v2_cat_1(A)
& v1_cat_5(A) ) ) ) ).
fof(rc3_cat_5,axiom,
? [A] :
( l1_cat_1(A)
& v1_cat_1(A)
& v2_cat_1(A)
& v1_cat_5(A)
& v3_cat_5(A) ) ).
fof(cc2_cat_5,axiom,
! [A] :
( ( v2_cat_1(A)
& v3_cat_5(A)
& l1_cat_1(A) )
=> ! [B] :
( m3_cat_2(B,A)
=> v3_cat_5(B) ) ) ).
fof(rc4_cat_5,axiom,
? [A] :
( l1_cat_1(A)
& v1_cat_1(A)
& v2_cat_1(A)
& v1_cat_5(A)
& v3_cat_5(A)
& v4_cat_5(A) ) ).
fof(fc1_cat_5,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& m1_subset_1(B,u1_cat_1(A)) )
=> ~ v1_xboole_0(k12_cat_5(A,B)) ) ).
fof(fc2_cat_5,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& m1_subset_1(B,u1_cat_1(A)) )
=> ~ v1_xboole_0(k13_cat_5(A,B)) ) ).
fof(t1_cat_5,axiom,
! [A] :
( l1_cat_1(A)
=> ! [B] :
( l1_cat_1(B)
=> ( ( g1_cat_1(u1_cat_1(A),u2_cat_1(A),u3_cat_1(A),u4_cat_1(A),u5_cat_1(A),u6_cat_1(A)) = g1_cat_1(u1_cat_1(B),u2_cat_1(B),u3_cat_1(B),u4_cat_1(B),u5_cat_1(B),u6_cat_1(B))
& v2_cat_1(A) )
=> v2_cat_1(B) ) ) ) ).
fof(d1_cat_5,axiom,
! [A] :
( l1_cat_1(A)
=> ( v1_cat_5(A)
<=> ! [B] :
( m1_subset_1(B,u2_cat_1(A))
=> ? [C] : B = k4_tarski(k1_domain_1(u1_cat_1(A),u1_cat_1(A),k2_cat_1(A,B),k3_cat_1(A,B)),C) ) ) ) ).
fof(t2_cat_5,axiom,
! [A] :
( ( v1_cat_5(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u2_cat_1(A))
=> ( k2_cat_1(A,B) = k17_mcart_1(B)
& k3_cat_1(A,B) = k18_mcart_1(B)
& B = k4_tarski(k1_domain_1(u1_cat_1(A),u1_cat_1(A),k2_cat_1(A,B),k3_cat_1(A,B)),k2_mcart_1(B)) ) ) ) ).
fof(t3_cat_5,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m3_cat_2(B,A)
=> ( m3_cat_2(A,B)
=> g1_cat_1(u1_cat_1(A),u2_cat_1(A),u3_cat_1(A),u4_cat_1(A),u5_cat_1(A),u6_cat_1(A)) = g1_cat_1(u1_cat_1(B),u2_cat_1(B),u3_cat_1(B),u4_cat_1(B),u5_cat_1(B),u6_cat_1(B)) ) ) ) ).
fof(t4_cat_5,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m3_cat_2(B,A)
=> ! [C] :
( m3_cat_2(C,B)
=> m3_cat_2(C,A) ) ) ) ).
fof(d2_cat_5,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ~ ( ? [C] :
( v2_cat_1(C)
& l1_cat_1(C)
& m3_cat_2(A,C)
& m3_cat_2(B,C) )
& ? [C] :
( m1_subset_1(C,u1_cat_1(A))
& m1_subset_1(C,u1_cat_1(B)) )
& ~ ! [C] :
( ( v1_cat_1(C)
& v2_cat_1(C)
& l1_cat_1(C) )
=> ( C = k6_cat_5(A,B)
<=> ( u1_cat_1(C) = k3_xboole_0(u1_cat_1(A),u1_cat_1(B))
& u2_cat_1(C) = k3_xboole_0(u2_cat_1(A),u2_cat_1(B))
& u3_cat_1(C) = k2_partfun1(u2_cat_1(A),u1_cat_1(A),u3_cat_1(A),u2_cat_1(B))
& u4_cat_1(C) = k2_partfun1(u2_cat_1(A),u1_cat_1(A),u4_cat_1(A),u2_cat_1(B))
& u5_cat_1(C) = k1_realset1(u5_cat_1(A),u2_cat_1(B))
& u6_cat_1(C) = k2_partfun1(u1_cat_1(A),u2_cat_1(A),u6_cat_1(A),u1_cat_1(B)) ) ) ) ) ) ) ).
fof(t5_cat_5,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m3_cat_2(B,A)
=> ! [C] :
( m3_cat_2(C,A)
=> ( ~ r2_subset_1(u1_cat_1(B),u1_cat_1(C))
=> k6_cat_5(B,C) = k6_cat_5(C,B) ) ) ) ) ).
fof(t6_cat_5,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m3_cat_2(B,A)
=> ! [C] :
( m3_cat_2(C,A)
=> ( ~ r2_subset_1(u1_cat_1(B),u1_cat_1(C))
=> ( m3_cat_2(k6_cat_5(B,C),B)
& m3_cat_2(k6_cat_5(B,C),C) ) ) ) ) ) ).
fof(d3_cat_5,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m2_cat_1(C,A,B)
=> ! [D] :
( ( v1_cat_1(D)
& m3_cat_2(D,B) )
=> ( D = k7_cat_5(A,B,C)
<=> ( u1_cat_1(D) = k2_relat_1(k12_cat_1(A,B,C))
& r1_tarski(k2_relat_1(C),u2_cat_1(D))
& ! [E] :
( m3_cat_2(E,B)
=> ( ( u1_cat_1(E) = k2_relat_1(k12_cat_1(A,B,C))
& r1_tarski(k2_relat_1(C),u2_cat_1(E)) )
=> m3_cat_2(D,E) ) ) ) ) ) ) ) ) ).
fof(t7_cat_5,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m3_cat_2(C,B)
=> ! [D] :
( m2_cat_1(D,A,B)
=> ( r1_tarski(k2_relat_1(D),u2_cat_1(C))
=> m2_cat_1(D,A,C) ) ) ) ) ) ).
fof(t8_cat_5,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m2_cat_1(C,A,B)
=> m2_cat_1(C,A,k7_cat_5(A,B,C)) ) ) ) ).
fof(t9_cat_5,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m3_cat_2(C,B)
=> ! [D] :
( m2_cat_1(D,A,C)
=> ! [E] :
( m2_cat_1(E,A,B)
=> ( D = E
=> k7_cat_5(A,C,D) = k7_cat_5(A,B,E) ) ) ) ) ) ) ).
fof(d4_cat_5,axiom,
! [A] :
( v2_cat_5(A)
<=> ! [B] :
( r2_hidden(B,A)
=> ( v2_cat_1(B)
& l1_cat_1(B) ) ) ) ).
fof(d5_cat_5,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( v2_cat_5(A)
<=> ! [B] :
( m1_subset_1(B,A)
=> ( v2_cat_1(B)
& l1_cat_1(B) ) ) ) ) ).
fof(d6_cat_5,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ( v3_cat_5(A)
<=> ( v2_cat_5(u1_cat_1(A))
& ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> ! [C] :
( ( v2_cat_1(C)
& l1_cat_1(C) )
=> ( B = C
=> k10_cat_1(A,B) = k4_tarski(k4_tarski(C,C),k15_cat_1(C)) ) ) )
& ! [B] :
( m1_subset_1(B,u2_cat_1(A))
=> ! [C] :
( ( v2_cat_1(C)
& l1_cat_1(C) )
=> ! [D] :
( ( v2_cat_1(D)
& l1_cat_1(D) )
=> ~ ( C = k2_cat_1(A,B)
& D = k3_cat_1(A,B)
& ! [E] :
( m2_cat_1(E,C,D)
=> B != k4_tarski(k4_tarski(C,D),E) ) ) ) ) )
& ! [B] :
( m1_subset_1(B,u2_cat_1(A))
=> ! [C] :
( m1_subset_1(C,u2_cat_1(A))
=> ! [D] :
( ( v2_cat_1(D)
& l1_cat_1(D) )
=> ! [E] :
( ( v2_cat_1(E)
& l1_cat_1(E) )
=> ! [F] :
( ( v2_cat_1(F)
& l1_cat_1(F) )
=> ! [G] :
( m2_cat_1(G,D,E)
=> ! [H] :
( m2_cat_1(H,E,F)
=> ( ( B = k4_tarski(k4_tarski(D,E),G)
& C = k4_tarski(k4_tarski(E,F),H) )
=> k4_cat_1(A,B,C) = k4_tarski(k4_tarski(D,F),k14_cat_1(D,E,F,G,H)) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t10_cat_5,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ( ( g1_cat_1(u1_cat_1(A),u2_cat_1(A),u3_cat_1(A),u4_cat_1(A),u5_cat_1(A),u6_cat_1(A)) = g1_cat_1(u1_cat_1(B),u2_cat_1(B),u3_cat_1(B),u4_cat_1(B),u5_cat_1(B),u6_cat_1(B))
& v3_cat_5(A) )
=> v3_cat_5(B) ) ) ) ).
fof(t11_cat_5,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> v3_cat_5(k8_cat_1(A,k4_tarski(k4_tarski(A,A),k15_cat_1(A)))) ) ).
fof(t12_cat_5,axiom,
! [A] :
( ( v2_cat_1(A)
& v3_cat_5(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> ( v2_cat_1(B)
& l1_cat_1(B) ) ) ) ).
fof(t13_cat_5,axiom,
! [A] :
( ( v2_cat_1(A)
& v3_cat_5(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u2_cat_1(A))
=> ( k2_cat_1(A,B) = k4_cat_5(A,B)
& k3_cat_1(A,B) = k5_cat_5(A,B) ) ) ) ).
fof(t14_cat_5,axiom,
! [A] :
( ( v2_cat_1(A)
& v3_cat_5(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& v3_cat_5(B)
& l1_cat_1(B) )
=> ( ( u1_cat_1(A) = u1_cat_1(B)
& u2_cat_1(A) = u2_cat_1(B) )
=> g1_cat_1(u1_cat_1(A),u2_cat_1(A),u3_cat_1(A),u4_cat_1(A),u5_cat_1(A),u6_cat_1(A)) = g1_cat_1(u1_cat_1(B),u2_cat_1(B),u3_cat_1(B),u4_cat_1(B),u5_cat_1(B),u6_cat_1(B)) ) ) ) ).
fof(t15_cat_5,axiom,
! [A] :
( ( v2_cat_1(A)
& v3_cat_5(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& v3_cat_5(B)
& l1_cat_1(B) )
=> ( r1_tarski(u2_cat_1(A),u2_cat_1(B))
=> m3_cat_2(A,B) ) ) ) ).
fof(d7_cat_5,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> k10_cat_5(A) = A ) ).
fof(t16_cat_5,axiom,
! [A] :
( ( v2_cat_1(A)
& v3_cat_5(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> k10_cat_5(B) = B ) ) ).
fof(t17_cat_5,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v2_cat_5(A) )
=> ! [B] :
( ~ v1_xboole_0(B)
=> ~ ( ! [C] :
( m1_cat_5(C,A)
=> ! [D] :
( m1_cat_5(D,A)
=> ! [E] :
( m1_cat_5(E,A)
=> ! [F] :
( m2_cat_1(F,C,D)
=> ! [G] :
( m2_cat_1(G,D,E)
=> ( ( r2_hidden(F,B)
& r2_hidden(G,B) )
=> r2_hidden(k14_cat_1(C,D,E,F,G),B) ) ) ) ) ) )
& ! [C] :
( m1_cat_5(C,A)
=> r2_hidden(k15_cat_1(C),B) )
& ! [C] :
( ( v1_cat_1(C)
& v2_cat_1(C)
& v3_cat_5(C)
& l1_cat_1(C) )
=> ~ ( u1_cat_1(C) = A
& ! [D] :
( m1_cat_5(D,A)
=> ! [E] :
( m1_cat_5(E,A)
=> ! [F] :
( m2_cat_1(F,D,E)
=> ( m1_subset_1(k1_domain_1(k2_zfmisc_1(A,A),k1_zfmisc_1(k2_zfmisc_1(u2_cat_1(D),u2_cat_1(E))),k1_domain_1(A,A,D,E),F),u2_cat_1(C))
<=> r2_hidden(F,B) ) ) ) ) ) ) ) ) ) ).
fof(t18_cat_5,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v2_cat_5(A) )
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_cat_1(C)
& v2_cat_1(C)
& v3_cat_5(C)
& l1_cat_1(C) )
=> ! [D] :
( ( v1_cat_1(D)
& v2_cat_1(D)
& v3_cat_5(D)
& l1_cat_1(D) )
=> ( ( u1_cat_1(C) = A
& ! [E] :
( m1_cat_5(E,A)
=> ! [F] :
( m1_cat_5(F,A)
=> ! [G] :
( m2_cat_1(G,E,F)
=> ( m1_subset_1(k1_domain_1(k2_zfmisc_1(A,A),k1_zfmisc_1(k2_zfmisc_1(u2_cat_1(E),u2_cat_1(F))),k1_domain_1(A,A,E,F),G),u2_cat_1(C))
<=> r2_hidden(G,B) ) ) ) )
& u1_cat_1(D) = A
& ! [E] :
( m1_cat_5(E,A)
=> ! [F] :
( m1_cat_5(F,A)
=> ! [G] :
( m2_cat_1(G,E,F)
=> ( m1_subset_1(k1_domain_1(k2_zfmisc_1(A,A),k1_zfmisc_1(k2_zfmisc_1(u2_cat_1(E),u2_cat_1(F))),k1_domain_1(A,A,E,F),G),u2_cat_1(D))
<=> r2_hidden(G,B) ) ) ) ) )
=> C = D ) ) ) ) ) ).
fof(d8_cat_5,axiom,
! [A] :
( ( v2_cat_1(A)
& v3_cat_5(A)
& l1_cat_1(A) )
=> ( v4_cat_5(A)
<=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( ( v2_cat_1(C)
& l1_cat_1(C) )
=> ( ( m1_subset_1(B,u1_cat_1(A))
& m1_subset_1(C,u1_cat_1(A)) )
=> ! [D] :
( m2_cat_1(D,B,C)
=> m1_subset_1(k4_tarski(k4_tarski(B,C),D),u2_cat_1(A)) ) ) ) ) ) ) ).
fof(t19_cat_5,axiom,
! [A] :
( ( v2_cat_1(A)
& v3_cat_5(A)
& v4_cat_5(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& v3_cat_5(B)
& v4_cat_5(B)
& l1_cat_1(B) )
=> ( u1_cat_1(A) = u1_cat_1(B)
=> g1_cat_1(u1_cat_1(A),u2_cat_1(A),u3_cat_1(A),u4_cat_1(A),u5_cat_1(A),u6_cat_1(A)) = g1_cat_1(u1_cat_1(B),u2_cat_1(B),u3_cat_1(B),u4_cat_1(B),u5_cat_1(B),u6_cat_1(B)) ) ) ) ).
fof(t20_cat_5,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v2_cat_5(A) )
=> ? [B] :
( v1_cat_1(B)
& v2_cat_1(B)
& v3_cat_5(B)
& v4_cat_5(B)
& l1_cat_1(B)
& u1_cat_1(B) = A ) ) ).
fof(t21_cat_5,axiom,
! [A] :
( ( v2_cat_1(A)
& v3_cat_5(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& v3_cat_5(B)
& v4_cat_5(B)
& l1_cat_1(B) )
=> ( r1_tarski(u1_cat_1(A),u1_cat_1(B))
=> m3_cat_2(A,B) ) ) ) ).
fof(t22_cat_5,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& v3_cat_5(B)
& l1_cat_1(B) )
=> ! [C] :
( ( v2_cat_1(C)
& v3_cat_5(C)
& l1_cat_1(C) )
=> ! [D] :
( m2_cat_1(D,A,B)
=> ! [E] :
( m2_cat_1(E,A,C)
=> ( D = E
=> k7_cat_5(A,B,D) = k7_cat_5(A,C,E) ) ) ) ) ) ) ).
fof(d9_cat_5,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> k12_cat_5(A,B) = k3_funct_2(u2_cat_1(A),u1_cat_1(A),u4_cat_1(A),k6_domain_1(u1_cat_1(A),B)) ) ) ).
fof(d10_cat_5,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> k13_cat_5(A,B) = k3_funct_2(u2_cat_1(A),u1_cat_1(A),u3_cat_1(A),k6_domain_1(u1_cat_1(A),B)) ) ) ).
fof(t23_cat_5,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> ! [C] :
( m1_subset_1(C,u2_cat_1(A))
=> ( r2_hidden(C,k12_cat_5(A,B))
<=> k3_cat_1(A,C) = B ) ) ) ) ).
fof(t24_cat_5,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> ! [C] :
( m1_subset_1(C,u2_cat_1(A))
=> ( r2_hidden(C,k13_cat_5(A,B))
<=> k2_cat_1(A,C) = B ) ) ) ) ).
fof(t25_cat_5,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> ! [C] :
( m1_subset_1(C,u1_cat_1(A))
=> k6_cat_1(A,B,C) = k5_subset_1(u2_cat_1(A),k13_cat_5(A,B),k12_cat_5(A,C)) ) ) ) ).
fof(t26_cat_5,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u2_cat_1(A))
=> ( r2_hidden(B,k13_cat_5(A,k2_cat_1(A,B)))
& r2_hidden(B,k12_cat_5(A,k3_cat_1(A,B))) ) ) ) ).
fof(t27_cat_5,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u2_cat_1(A))
=> ! [C] :
( m2_subset_1(C,u2_cat_1(A),k12_cat_5(A,k2_cat_1(A,B)))
=> r2_hidden(k4_cat_1(A,C,B),k12_cat_5(A,k3_cat_1(A,B))) ) ) ) ).
fof(t28_cat_5,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u2_cat_1(A))
=> ! [C] :
( m2_subset_1(C,u2_cat_1(A),k13_cat_5(A,k3_cat_1(A,B)))
=> r2_hidden(k4_cat_1(A,B,C),k13_cat_5(A,k2_cat_1(A,B))) ) ) ) ).
fof(d11_cat_5,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> ! [C] :
( ( v1_cat_1(C)
& v2_cat_1(C)
& v1_cat_5(C)
& l1_cat_1(C) )
=> ( C = k14_cat_5(A,B)
<=> ( u1_cat_1(C) = k12_cat_5(A,B)
& ! [D] :
( m2_subset_1(D,u2_cat_1(A),k12_cat_5(A,B))
=> ! [E] :
( m2_subset_1(E,u2_cat_1(A),k12_cat_5(A,B))
=> ! [F] :
( m1_subset_1(F,u2_cat_1(A))
=> ( ( k2_cat_1(A,E) = k3_cat_1(A,F)
& D = k4_cat_1(A,F,E) )
=> m1_subset_1(k13_cat_2(k11_cat_2(A,A),A,k13_cat_2(A,A,D,E),F),u2_cat_1(C)) ) ) ) )
& ! [D] :
( m1_subset_1(D,u2_cat_1(C))
=> ? [E] :
( m2_subset_1(E,u2_cat_1(A),k12_cat_5(A,B))
& ? [F] :
( m2_subset_1(F,u2_cat_1(A),k12_cat_5(A,B))
& ? [G] :
( m1_subset_1(G,u2_cat_1(A))
& D = k13_cat_2(k11_cat_2(A,A),A,k13_cat_2(A,A,E,F),G)
& k2_cat_1(A,F) = k3_cat_1(A,G)
& E = k4_cat_1(A,G,F) ) ) ) )
& ! [D] :
( m1_subset_1(D,u2_cat_1(C))
=> ! [E] :
( m1_subset_1(E,u2_cat_1(C))
=> ! [F] :
( m2_subset_1(F,u2_cat_1(A),k12_cat_5(A,B))
=> ! [G] :
( m2_subset_1(G,u2_cat_1(A),k12_cat_5(A,B))
=> ! [H] :
( m2_subset_1(H,u2_cat_1(A),k12_cat_5(A,B))
=> ! [I] :
( m1_subset_1(I,u2_cat_1(A))
=> ! [J] :
( m1_subset_1(J,u2_cat_1(A))
=> ( ( D = k13_cat_2(k11_cat_2(A,A),A,k13_cat_2(A,A,F,G),I)
& E = k13_cat_2(k11_cat_2(A,A),A,k13_cat_2(A,A,G,H),J) )
=> k4_cat_1(C,D,E) = k13_cat_2(k11_cat_2(A,A),A,k13_cat_2(A,A,F,H),k4_cat_1(A,I,J)) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d12_cat_5,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> ! [C] :
( ( v1_cat_1(C)
& v2_cat_1(C)
& v1_cat_5(C)
& l1_cat_1(C) )
=> ( C = k15_cat_5(A,B)
<=> ( u1_cat_1(C) = k13_cat_5(A,B)
& ! [D] :
( m2_subset_1(D,u2_cat_1(A),k13_cat_5(A,B))
=> ! [E] :
( m2_subset_1(E,u2_cat_1(A),k13_cat_5(A,B))
=> ! [F] :
( m1_subset_1(F,u2_cat_1(A))
=> ( ( k2_cat_1(A,F) = k3_cat_1(A,D)
& k4_cat_1(A,D,F) = E )
=> m1_subset_1(k13_cat_2(k11_cat_2(A,A),A,k13_cat_2(A,A,D,E),F),u2_cat_1(C)) ) ) ) )
& ! [D] :
( m1_subset_1(D,u2_cat_1(C))
=> ? [E] :
( m2_subset_1(E,u2_cat_1(A),k13_cat_5(A,B))
& ? [F] :
( m2_subset_1(F,u2_cat_1(A),k13_cat_5(A,B))
& ? [G] :
( m1_subset_1(G,u2_cat_1(A))
& D = k13_cat_2(k11_cat_2(A,A),A,k13_cat_2(A,A,E,F),G)
& k2_cat_1(A,G) = k3_cat_1(A,E)
& k4_cat_1(A,E,G) = F ) ) ) )
& ! [D] :
( m1_subset_1(D,u2_cat_1(C))
=> ! [E] :
( m1_subset_1(E,u2_cat_1(C))
=> ! [F] :
( m2_subset_1(F,u2_cat_1(A),k13_cat_5(A,B))
=> ! [G] :
( m2_subset_1(G,u2_cat_1(A),k13_cat_5(A,B))
=> ! [H] :
( m2_subset_1(H,u2_cat_1(A),k13_cat_5(A,B))
=> ! [I] :
( m1_subset_1(I,u2_cat_1(A))
=> ! [J] :
( m1_subset_1(J,u2_cat_1(A))
=> ( ( D = k13_cat_2(k11_cat_2(A,A),A,k13_cat_2(A,A,F,G),I)
& E = k13_cat_2(k11_cat_2(A,A),A,k13_cat_2(A,A,G,H),J) )
=> k4_cat_1(C,D,E) = k13_cat_2(k11_cat_2(A,A),A,k13_cat_2(A,A,F,H),k4_cat_1(A,I,J)) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t29_cat_5,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> ! [C] :
( m1_subset_1(C,u2_cat_1(k14_cat_5(A,B)))
=> ( C = k13_cat_2(k11_cat_2(A,A),A,k13_cat_2(A,A,k17_cat_5(A,B,C),k18_cat_5(A,B,C)),k16_cat_5(A,B,C))
& k2_cat_1(A,k18_cat_5(A,B,C)) = k3_cat_1(A,k16_cat_5(A,B,C))
& k17_cat_5(A,B,C) = k4_cat_1(A,k16_cat_5(A,B,C),k18_cat_5(A,B,C))
& k2_cat_1(k14_cat_5(A,B),C) = k17_cat_5(A,B,C)
& k3_cat_1(k14_cat_5(A,B),C) = k18_cat_5(A,B,C) ) ) ) ) ).
fof(t30_cat_5,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> ! [C] :
( m2_subset_1(C,u2_cat_1(A),k12_cat_5(A,B))
=> ! [D] :
( m1_subset_1(D,u1_cat_1(k14_cat_5(A,B)))
=> ( D = C
=> k10_cat_1(k14_cat_5(A,B),D) = k1_domain_1(u1_cat_1(k11_cat_2(k14_cat_5(A,B),k14_cat_5(A,B))),u2_cat_1(A),k12_cat_2(k14_cat_5(A,B),k14_cat_5(A,B),D,D),k10_cat_1(A,k2_cat_1(A,C))) ) ) ) ) ) ).
fof(t31_cat_5,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> ! [C] :
( m1_subset_1(C,u2_cat_1(k15_cat_5(A,B)))
=> ( C = k13_cat_2(k11_cat_2(A,A),A,k13_cat_2(A,A,k20_cat_5(A,B,C),k21_cat_5(A,B,C)),k19_cat_5(A,B,C))
& k2_cat_1(A,k19_cat_5(A,B,C)) = k3_cat_1(A,k20_cat_5(A,B,C))
& k4_cat_1(A,k20_cat_5(A,B,C),k19_cat_5(A,B,C)) = k21_cat_5(A,B,C)
& k2_cat_1(k15_cat_5(A,B),C) = k20_cat_5(A,B,C)
& k3_cat_1(k15_cat_5(A,B),C) = k21_cat_5(A,B,C) ) ) ) ) ).
fof(t32_cat_5,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> ! [C] :
( m2_subset_1(C,u2_cat_1(A),k13_cat_5(A,B))
=> ! [D] :
( m1_subset_1(D,u1_cat_1(k15_cat_5(A,B)))
=> ( D = C
=> k10_cat_1(k15_cat_5(A,B),D) = k1_domain_1(u1_cat_1(k11_cat_2(k15_cat_5(A,B),k15_cat_5(A,B))),u2_cat_1(A),k12_cat_2(k15_cat_5(A,B),k15_cat_5(A,B),D,D),k10_cat_1(A,k3_cat_1(A,C))) ) ) ) ) ) ).
fof(d13_cat_5,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u2_cat_1(A))
=> ! [C] :
( m2_cat_1(C,k14_cat_5(A,k2_cat_1(A,B)),k14_cat_5(A,k3_cat_1(A,B)))
=> ( C = k22_cat_5(A,B)
<=> ! [D] :
( m1_subset_1(D,u2_cat_1(k14_cat_5(A,k2_cat_1(A,B))))
=> k8_funct_2(u2_cat_1(k14_cat_5(A,k2_cat_1(A,B))),u2_cat_1(k14_cat_5(A,k3_cat_1(A,B))),C,D) = k13_cat_2(k11_cat_2(A,A),A,k13_cat_2(A,A,k4_cat_1(A,k17_cat_5(A,k2_cat_1(A,B),D),B),k4_cat_1(A,k18_cat_5(A,k2_cat_1(A,B),D),B)),k16_cat_5(A,k2_cat_1(A,B),D)) ) ) ) ) ) ).
fof(d14_cat_5,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u2_cat_1(A))
=> ! [C] :
( m2_cat_1(C,k15_cat_5(A,k3_cat_1(A,B)),k15_cat_5(A,k2_cat_1(A,B)))
=> ( C = k23_cat_5(A,B)
<=> ! [D] :
( m1_subset_1(D,u2_cat_1(k15_cat_5(A,k3_cat_1(A,B))))
=> k8_funct_2(u2_cat_1(k15_cat_5(A,k3_cat_1(A,B))),u2_cat_1(k15_cat_5(A,k2_cat_1(A,B))),C,D) = k13_cat_2(k11_cat_2(A,A),A,k13_cat_2(A,A,k4_cat_1(A,B,k20_cat_5(A,k3_cat_1(A,B),D)),k4_cat_1(A,B,k21_cat_5(A,k3_cat_1(A,B),D))),k19_cat_5(A,k3_cat_1(A,B),D)) ) ) ) ) ) ).
fof(t33_cat_5,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u2_cat_1(A))
=> ! [C] :
( m1_subset_1(C,u2_cat_1(A))
=> ( k2_cat_1(A,C) = k3_cat_1(A,B)
=> k22_cat_5(A,k4_cat_1(A,B,C)) = k1_partfun1(u2_cat_1(k14_cat_5(A,k2_cat_1(A,B))),u2_cat_1(k14_cat_5(A,k3_cat_1(A,B))),u2_cat_1(k14_cat_5(A,k2_cat_1(A,C))),u2_cat_1(k14_cat_5(A,k3_cat_1(A,C))),k22_cat_5(A,B),k22_cat_5(A,C)) ) ) ) ) ).
fof(t34_cat_5,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u2_cat_1(A))
=> ! [C] :
( m1_subset_1(C,u2_cat_1(A))
=> ( k2_cat_1(A,C) = k3_cat_1(A,B)
=> k23_cat_5(A,k4_cat_1(A,B,C)) = k1_partfun1(u2_cat_1(k15_cat_5(A,k3_cat_1(A,C))),u2_cat_1(k15_cat_5(A,k2_cat_1(A,C))),u2_cat_1(k15_cat_5(A,k3_cat_1(A,B))),u2_cat_1(k15_cat_5(A,k2_cat_1(A,B))),k23_cat_5(A,C),k23_cat_5(A,B)) ) ) ) ) ).
fof(s1_cat_5,axiom,
( ( ! [A] :
( m1_subset_1(A,f1_s1_cat_5)
=> ! [B] :
( m1_subset_1(B,f1_s1_cat_5)
=> ! [C] :
( m1_subset_1(C,f1_s1_cat_5)
=> ! [D] :
( m1_subset_1(D,f2_s1_cat_5)
=> ! [E] :
( m1_subset_1(E,f2_s1_cat_5)
=> ( ( p1_s1_cat_5(A,B,D)
& p1_s1_cat_5(B,C,E) )
=> ( r2_hidden(f3_s1_cat_5(E,D),f2_s1_cat_5)
& p1_s1_cat_5(A,C,f3_s1_cat_5(E,D)) ) ) ) ) ) ) )
& ! [A] :
( m1_subset_1(A,f1_s1_cat_5)
=> ? [B] :
( m1_subset_1(B,f2_s1_cat_5)
& p1_s1_cat_5(A,A,B)
& ! [C] :
( m1_subset_1(C,f1_s1_cat_5)
=> ! [D] :
( m1_subset_1(D,f2_s1_cat_5)
=> ( ( p1_s1_cat_5(A,C,D)
=> f3_s1_cat_5(D,B) = D )
& ( p1_s1_cat_5(C,A,D)
=> f3_s1_cat_5(B,D) = D ) ) ) ) ) )
& ! [A] :
( m1_subset_1(A,f1_s1_cat_5)
=> ! [B] :
( m1_subset_1(B,f1_s1_cat_5)
=> ! [C] :
( m1_subset_1(C,f1_s1_cat_5)
=> ! [D] :
( m1_subset_1(D,f1_s1_cat_5)
=> ! [E] :
( m1_subset_1(E,f2_s1_cat_5)
=> ! [F] :
( m1_subset_1(F,f2_s1_cat_5)
=> ! [G] :
( m1_subset_1(G,f2_s1_cat_5)
=> ( ( p1_s1_cat_5(A,B,E)
& p1_s1_cat_5(B,C,F)
& p1_s1_cat_5(C,D,G) )
=> f3_s1_cat_5(G,f3_s1_cat_5(F,E)) = f3_s1_cat_5(f3_s1_cat_5(G,F),E) ) ) ) ) ) ) ) ) )
=> ? [A] :
( v1_cat_1(A)
& v2_cat_1(A)
& v1_cat_5(A)
& l1_cat_1(A)
& u1_cat_1(A) = f1_s1_cat_5
& ! [B] :
( m1_subset_1(B,f1_s1_cat_5)
=> ! [C] :
( m1_subset_1(C,f1_s1_cat_5)
=> ! [D] :
( m1_subset_1(D,f2_s1_cat_5)
=> ( p1_s1_cat_5(B,C,D)
=> m1_subset_1(k1_domain_1(k2_zfmisc_1(f1_s1_cat_5,f1_s1_cat_5),f2_s1_cat_5,k1_domain_1(f1_s1_cat_5,f1_s1_cat_5,B,C),D),u2_cat_1(A)) ) ) ) )
& ! [B] :
( m1_subset_1(B,u2_cat_1(A))
=> ? [C] :
( m1_subset_1(C,f1_s1_cat_5)
& ? [D] :
( m1_subset_1(D,f1_s1_cat_5)
& ? [E] :
( m1_subset_1(E,f2_s1_cat_5)
& B = k1_domain_1(k2_zfmisc_1(f1_s1_cat_5,f1_s1_cat_5),f2_s1_cat_5,k1_domain_1(f1_s1_cat_5,f1_s1_cat_5,C,D),E)
& p1_s1_cat_5(C,D,E) ) ) ) )
& ! [B] :
( m1_subset_1(B,u2_cat_1(A))
=> ! [C] :
( m1_subset_1(C,u2_cat_1(A))
=> ! [D] :
( m1_subset_1(D,f1_s1_cat_5)
=> ! [E] :
( m1_subset_1(E,f1_s1_cat_5)
=> ! [F] :
( m1_subset_1(F,f1_s1_cat_5)
=> ! [G] :
( m1_subset_1(G,f2_s1_cat_5)
=> ! [H] :
( m1_subset_1(H,f2_s1_cat_5)
=> ( ( B = k1_domain_1(k2_zfmisc_1(f1_s1_cat_5,f1_s1_cat_5),f2_s1_cat_5,k1_domain_1(f1_s1_cat_5,f1_s1_cat_5,D,E),G)
& C = k1_domain_1(k2_zfmisc_1(f1_s1_cat_5,f1_s1_cat_5),f2_s1_cat_5,k1_domain_1(f1_s1_cat_5,f1_s1_cat_5,E,F),H) )
=> k4_cat_1(A,B,C) = k4_tarski(k1_domain_1(f1_s1_cat_5,f1_s1_cat_5,D,F),f3_s1_cat_5(H,G)) ) ) ) ) ) ) ) ) ) ) ).
fof(s2_cat_5,axiom,
( ! [A] :
( m1_subset_1(A,f1_s2_cat_5)
=> ? [B] :
( m1_subset_1(B,f2_s2_cat_5)
& p1_s2_cat_5(A,A,B)
& ! [C] :
( m1_subset_1(C,f1_s2_cat_5)
=> ! [D] :
( m1_subset_1(D,f2_s2_cat_5)
=> ( ( p1_s2_cat_5(A,C,D)
=> f3_s2_cat_5(D,B) = D )
& ( p1_s2_cat_5(C,A,D)
=> f3_s2_cat_5(B,D) = D ) ) ) ) ) )
=> ! [A] :
( ( v1_cat_1(A)
& v2_cat_1(A)
& v1_cat_5(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v1_cat_1(B)
& v2_cat_1(B)
& v1_cat_5(B)
& l1_cat_1(B) )
=> ( ( u1_cat_1(A) = f1_s2_cat_5
& ! [C] :
( m1_subset_1(C,f1_s2_cat_5)
=> ! [D] :
( m1_subset_1(D,f1_s2_cat_5)
=> ! [E] :
( m1_subset_1(E,f2_s2_cat_5)
=> ( p1_s2_cat_5(C,D,E)
=> m1_subset_1(k1_domain_1(k2_zfmisc_1(f1_s2_cat_5,f1_s2_cat_5),f2_s2_cat_5,k1_domain_1(f1_s2_cat_5,f1_s2_cat_5,C,D),E),u2_cat_1(A)) ) ) ) )
& ! [C] :
( m1_subset_1(C,u2_cat_1(A))
=> ? [D] :
( m1_subset_1(D,f1_s2_cat_5)
& ? [E] :
( m1_subset_1(E,f1_s2_cat_5)
& ? [F] :
( m1_subset_1(F,f2_s2_cat_5)
& C = k1_domain_1(k2_zfmisc_1(f1_s2_cat_5,f1_s2_cat_5),f2_s2_cat_5,k1_domain_1(f1_s2_cat_5,f1_s2_cat_5,D,E),F)
& p1_s2_cat_5(D,E,F) ) ) ) )
& ! [C] :
( m1_subset_1(C,u2_cat_1(A))
=> ! [D] :
( m1_subset_1(D,u2_cat_1(A))
=> ! [E] :
( m1_subset_1(E,f1_s2_cat_5)
=> ! [F] :
( m1_subset_1(F,f1_s2_cat_5)
=> ! [G] :
( m1_subset_1(G,f1_s2_cat_5)
=> ! [H] :
( m1_subset_1(H,f2_s2_cat_5)
=> ! [I] :
( m1_subset_1(I,f2_s2_cat_5)
=> ( ( C = k1_domain_1(k2_zfmisc_1(f1_s2_cat_5,f1_s2_cat_5),f2_s2_cat_5,k1_domain_1(f1_s2_cat_5,f1_s2_cat_5,E,F),H)
& D = k1_domain_1(k2_zfmisc_1(f1_s2_cat_5,f1_s2_cat_5),f2_s2_cat_5,k1_domain_1(f1_s2_cat_5,f1_s2_cat_5,F,G),I) )
=> k4_cat_1(A,C,D) = k4_tarski(k1_domain_1(f1_s2_cat_5,f1_s2_cat_5,E,G),f3_s2_cat_5(I,H)) ) ) ) ) ) ) ) )
& u1_cat_1(B) = f1_s2_cat_5
& ! [C] :
( m1_subset_1(C,f1_s2_cat_5)
=> ! [D] :
( m1_subset_1(D,f1_s2_cat_5)
=> ! [E] :
( m1_subset_1(E,f2_s2_cat_5)
=> ( p1_s2_cat_5(C,D,E)
=> m1_subset_1(k1_domain_1(k2_zfmisc_1(f1_s2_cat_5,f1_s2_cat_5),f2_s2_cat_5,k1_domain_1(f1_s2_cat_5,f1_s2_cat_5,C,D),E),u2_cat_1(B)) ) ) ) )
& ! [C] :
( m1_subset_1(C,u2_cat_1(B))
=> ? [D] :
( m1_subset_1(D,f1_s2_cat_5)
& ? [E] :
( m1_subset_1(E,f1_s2_cat_5)
& ? [F] :
( m1_subset_1(F,f2_s2_cat_5)
& C = k1_domain_1(k2_zfmisc_1(f1_s2_cat_5,f1_s2_cat_5),f2_s2_cat_5,k1_domain_1(f1_s2_cat_5,f1_s2_cat_5,D,E),F)
& p1_s2_cat_5(D,E,F) ) ) ) )
& ! [C] :
( m1_subset_1(C,u2_cat_1(B))
=> ! [D] :
( m1_subset_1(D,u2_cat_1(B))
=> ! [E] :
( m1_subset_1(E,f1_s2_cat_5)
=> ! [F] :
( m1_subset_1(F,f1_s2_cat_5)
=> ! [G] :
( m1_subset_1(G,f1_s2_cat_5)
=> ! [H] :
( m1_subset_1(H,f2_s2_cat_5)
=> ! [I] :
( m1_subset_1(I,f2_s2_cat_5)
=> ( ( C = k1_domain_1(k2_zfmisc_1(f1_s2_cat_5,f1_s2_cat_5),f2_s2_cat_5,k1_domain_1(f1_s2_cat_5,f1_s2_cat_5,E,F),H)
& D = k1_domain_1(k2_zfmisc_1(f1_s2_cat_5,f1_s2_cat_5),f2_s2_cat_5,k1_domain_1(f1_s2_cat_5,f1_s2_cat_5,F,G),I) )
=> k4_cat_1(B,C,D) = k4_tarski(k1_domain_1(f1_s2_cat_5,f1_s2_cat_5,E,G),f3_s2_cat_5(I,H)) ) ) ) ) ) ) ) ) )
=> A = B ) ) ) ) ).
fof(s3_cat_5,axiom,
( ( ! [A] :
( m1_subset_1(A,u2_cat_1(f1_s3_cat_5))
=> ( m1_subset_1(f4_s3_cat_5(A),u2_cat_1(f2_s3_cat_5))
& ! [B] :
( m1_subset_1(B,u2_cat_1(f2_s3_cat_5))
=> ( B = f4_s3_cat_5(A)
=> ( k2_cat_1(f2_s3_cat_5,B) = f3_s3_cat_5(k2_cat_1(f1_s3_cat_5,A))
& k3_cat_1(f2_s3_cat_5,B) = f3_s3_cat_5(k3_cat_1(f1_s3_cat_5,A)) ) ) ) ) )
& ! [A] :
( m1_subset_1(A,u1_cat_1(f1_s3_cat_5))
=> f4_s3_cat_5(k10_cat_1(f1_s3_cat_5,A)) = k10_cat_1(f2_s3_cat_5,f3_s3_cat_5(A)) )
& ! [A] :
( m1_subset_1(A,u2_cat_1(f1_s3_cat_5))
=> ! [B] :
( m1_subset_1(B,u2_cat_1(f1_s3_cat_5))
=> ! [C] :
( m1_subset_1(C,u2_cat_1(f2_s3_cat_5))
=> ! [D] :
( m1_subset_1(D,u2_cat_1(f2_s3_cat_5))
=> ( ( C = f4_s3_cat_5(A)
& D = f4_s3_cat_5(B)
& k2_cat_1(f1_s3_cat_5,B) = k3_cat_1(f1_s3_cat_5,A) )
=> f4_s3_cat_5(k4_cat_1(f1_s3_cat_5,A,B)) = k4_cat_1(f2_s3_cat_5,C,D) ) ) ) ) ) )
=> ? [A] :
( m2_cat_1(A,f1_s3_cat_5,f2_s3_cat_5)
& ! [B] :
( m1_subset_1(B,u2_cat_1(f1_s3_cat_5))
=> k8_funct_2(u2_cat_1(f1_s3_cat_5),u2_cat_1(f2_s3_cat_5),A,B) = f4_s3_cat_5(B) ) ) ) ).
fof(dt_m1_cat_5,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v2_cat_5(A) )
=> ! [B] :
( m1_cat_5(B,A)
=> ( v2_cat_1(B)
& l1_cat_1(B) ) ) ) ).
fof(existence_m1_cat_5,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v2_cat_5(A) )
=> ? [B] : m1_cat_5(B,A) ) ).
fof(redefinition_m1_cat_5,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v2_cat_5(A) )
=> ! [B] :
( m1_cat_5(B,A)
<=> m1_subset_1(B,A) ) ) ).
fof(dt_k1_cat_5,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& m1_subset_1(D,k2_zfmisc_1(k2_zfmisc_1(A,B),C)) )
=> m1_subset_1(k1_cat_5(A,B,C,D),A) ) ).
fof(redefinition_k1_cat_5,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& m1_subset_1(D,k2_zfmisc_1(k2_zfmisc_1(A,B),C)) )
=> k1_cat_5(A,B,C,D) = k17_mcart_1(D) ) ).
fof(dt_k2_cat_5,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& m1_subset_1(D,k2_zfmisc_1(k2_zfmisc_1(A,B),C)) )
=> m1_subset_1(k2_cat_5(A,B,C,D),B) ) ).
fof(redefinition_k2_cat_5,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& m1_subset_1(D,k2_zfmisc_1(k2_zfmisc_1(A,B),C)) )
=> k2_cat_5(A,B,C,D) = k18_mcart_1(D) ) ).
fof(dt_k3_cat_5,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m1_subset_1(C,k2_zfmisc_1(A,B)) )
=> m1_subset_1(k3_cat_5(A,B,C),B) ) ).
fof(redefinition_k3_cat_5,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m1_subset_1(C,k2_zfmisc_1(A,B)) )
=> k3_cat_5(A,B,C) = k2_mcart_1(C) ) ).
fof(dt_k4_cat_5,axiom,
! [A,B] :
( ( v1_cat_5(A)
& l1_cat_1(A)
& m1_subset_1(B,u2_cat_1(A)) )
=> m1_subset_1(k4_cat_5(A,B),u1_cat_1(A)) ) ).
fof(redefinition_k4_cat_5,axiom,
! [A,B] :
( ( v1_cat_5(A)
& l1_cat_1(A)
& m1_subset_1(B,u2_cat_1(A)) )
=> k4_cat_5(A,B) = k17_mcart_1(B) ) ).
fof(dt_k5_cat_5,axiom,
! [A,B] :
( ( v1_cat_5(A)
& l1_cat_1(A)
& m1_subset_1(B,u2_cat_1(A)) )
=> m1_subset_1(k5_cat_5(A,B),u1_cat_1(A)) ) ).
fof(redefinition_k5_cat_5,axiom,
! [A,B] :
( ( v1_cat_5(A)
& l1_cat_1(A)
& m1_subset_1(B,u2_cat_1(A)) )
=> k5_cat_5(A,B) = k18_mcart_1(B) ) ).
fof(dt_k6_cat_5,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B) )
=> ( v1_cat_1(k6_cat_5(A,B))
& v2_cat_1(k6_cat_5(A,B))
& l1_cat_1(k6_cat_5(A,B)) ) ) ).
fof(dt_k7_cat_5,axiom,
! [A,B,C] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& m2_cat_1(C,A,B) )
=> ( v1_cat_1(k7_cat_5(A,B,C))
& m3_cat_2(k7_cat_5(A,B,C),B) ) ) ).
fof(dt_k8_cat_5,axiom,
! [A,B] :
( ( v2_cat_1(A)
& v3_cat_5(A)
& l1_cat_1(A)
& m1_subset_1(B,u2_cat_1(A)) )
=> ( v2_cat_1(k8_cat_5(A,B))
& l1_cat_1(k8_cat_5(A,B)) ) ) ).
fof(redefinition_k8_cat_5,axiom,
! [A,B] :
( ( v2_cat_1(A)
& v3_cat_5(A)
& l1_cat_1(A)
& m1_subset_1(B,u2_cat_1(A)) )
=> k8_cat_5(A,B) = k17_mcart_1(B) ) ).
fof(dt_k9_cat_5,axiom,
! [A,B] :
( ( v2_cat_1(A)
& v3_cat_5(A)
& l1_cat_1(A)
& m1_subset_1(B,u2_cat_1(A)) )
=> ( v2_cat_1(k9_cat_5(A,B))
& l1_cat_1(k9_cat_5(A,B)) ) ) ).
fof(redefinition_k9_cat_5,axiom,
! [A,B] :
( ( v2_cat_1(A)
& v3_cat_5(A)
& l1_cat_1(A)
& m1_subset_1(B,u2_cat_1(A)) )
=> k9_cat_5(A,B) = k18_mcart_1(B) ) ).
fof(dt_k10_cat_5,axiom,
! [A] :
( v2_cat_1(k10_cat_5(A))
& l1_cat_1(k10_cat_5(A)) ) ).
fof(dt_k11_cat_5,axiom,
! [A,B] :
( ( v2_cat_1(A)
& v3_cat_5(A)
& l1_cat_1(A)
& m1_subset_1(B,u2_cat_1(A)) )
=> m2_cat_1(k11_cat_5(A,B),k10_cat_5(k2_cat_1(A,B)),k10_cat_5(k3_cat_1(A,B))) ) ).
fof(redefinition_k11_cat_5,axiom,
! [A,B] :
( ( v2_cat_1(A)
& v3_cat_5(A)
& l1_cat_1(A)
& m1_subset_1(B,u2_cat_1(A)) )
=> k11_cat_5(A,B) = k2_mcart_1(B) ) ).
fof(dt_k12_cat_5,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& m1_subset_1(B,u1_cat_1(A)) )
=> m1_subset_1(k12_cat_5(A,B),k1_zfmisc_1(u2_cat_1(A))) ) ).
fof(dt_k13_cat_5,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& m1_subset_1(B,u1_cat_1(A)) )
=> m1_subset_1(k13_cat_5(A,B),k1_zfmisc_1(u2_cat_1(A))) ) ).
fof(dt_k14_cat_5,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& m1_subset_1(B,u1_cat_1(A)) )
=> ( v1_cat_1(k14_cat_5(A,B))
& v2_cat_1(k14_cat_5(A,B))
& v1_cat_5(k14_cat_5(A,B))
& l1_cat_1(k14_cat_5(A,B)) ) ) ).
fof(dt_k15_cat_5,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& m1_subset_1(B,u1_cat_1(A)) )
=> ( v1_cat_1(k15_cat_5(A,B))
& v2_cat_1(k15_cat_5(A,B))
& v1_cat_5(k15_cat_5(A,B))
& l1_cat_1(k15_cat_5(A,B)) ) ) ).
fof(dt_k16_cat_5,axiom,
! [A,B,C] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& m1_subset_1(B,u1_cat_1(A))
& m1_subset_1(C,u2_cat_1(k14_cat_5(A,B))) )
=> m1_subset_1(k16_cat_5(A,B,C),u2_cat_1(A)) ) ).
fof(redefinition_k16_cat_5,axiom,
! [A,B,C] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& m1_subset_1(B,u1_cat_1(A))
& m1_subset_1(C,u2_cat_1(k14_cat_5(A,B))) )
=> k16_cat_5(A,B,C) = k2_mcart_1(C) ) ).
fof(dt_k17_cat_5,axiom,
! [A,B,C] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& m1_subset_1(B,u1_cat_1(A))
& m1_subset_1(C,u2_cat_1(k14_cat_5(A,B))) )
=> m2_subset_1(k17_cat_5(A,B,C),u2_cat_1(A),k12_cat_5(A,B)) ) ).
fof(redefinition_k17_cat_5,axiom,
! [A,B,C] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& m1_subset_1(B,u1_cat_1(A))
& m1_subset_1(C,u2_cat_1(k14_cat_5(A,B))) )
=> k17_cat_5(A,B,C) = k17_mcart_1(C) ) ).
fof(dt_k18_cat_5,axiom,
! [A,B,C] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& m1_subset_1(B,u1_cat_1(A))
& m1_subset_1(C,u2_cat_1(k14_cat_5(A,B))) )
=> m2_subset_1(k18_cat_5(A,B,C),u2_cat_1(A),k12_cat_5(A,B)) ) ).
fof(redefinition_k18_cat_5,axiom,
! [A,B,C] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& m1_subset_1(B,u1_cat_1(A))
& m1_subset_1(C,u2_cat_1(k14_cat_5(A,B))) )
=> k18_cat_5(A,B,C) = k18_mcart_1(C) ) ).
fof(dt_k19_cat_5,axiom,
! [A,B,C] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& m1_subset_1(B,u1_cat_1(A))
& m1_subset_1(C,u2_cat_1(k15_cat_5(A,B))) )
=> m1_subset_1(k19_cat_5(A,B,C),u2_cat_1(A)) ) ).
fof(redefinition_k19_cat_5,axiom,
! [A,B,C] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& m1_subset_1(B,u1_cat_1(A))
& m1_subset_1(C,u2_cat_1(k15_cat_5(A,B))) )
=> k19_cat_5(A,B,C) = k2_mcart_1(C) ) ).
fof(dt_k20_cat_5,axiom,
! [A,B,C] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& m1_subset_1(B,u1_cat_1(A))
& m1_subset_1(C,u2_cat_1(k15_cat_5(A,B))) )
=> m2_subset_1(k20_cat_5(A,B,C),u2_cat_1(A),k13_cat_5(A,B)) ) ).
fof(redefinition_k20_cat_5,axiom,
! [A,B,C] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& m1_subset_1(B,u1_cat_1(A))
& m1_subset_1(C,u2_cat_1(k15_cat_5(A,B))) )
=> k20_cat_5(A,B,C) = k17_mcart_1(C) ) ).
fof(dt_k21_cat_5,axiom,
! [A,B,C] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& m1_subset_1(B,u1_cat_1(A))
& m1_subset_1(C,u2_cat_1(k15_cat_5(A,B))) )
=> m2_subset_1(k21_cat_5(A,B,C),u2_cat_1(A),k13_cat_5(A,B)) ) ).
fof(redefinition_k21_cat_5,axiom,
! [A,B,C] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& m1_subset_1(B,u1_cat_1(A))
& m1_subset_1(C,u2_cat_1(k15_cat_5(A,B))) )
=> k21_cat_5(A,B,C) = k18_mcart_1(C) ) ).
fof(dt_k22_cat_5,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& m1_subset_1(B,u2_cat_1(A)) )
=> m2_cat_1(k22_cat_5(A,B),k14_cat_5(A,k2_cat_1(A,B)),k14_cat_5(A,k3_cat_1(A,B))) ) ).
fof(dt_k23_cat_5,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& m1_subset_1(B,u2_cat_1(A)) )
=> m2_cat_1(k23_cat_5(A,B),k15_cat_5(A,k3_cat_1(A,B)),k15_cat_5(A,k2_cat_1(A,B))) ) ).
%------------------------------------------------------------------------------