SET007 Axioms: SET007+190.ax
%------------------------------------------------------------------------------
% File : SET007+190 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Miscellaneous Facts about 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_7 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 139 ( 10 unt; 0 def)
% Number of atoms : 849 ( 122 equ)
% Maximal formula atoms : 20 ( 6 avg)
% Number of connectives : 755 ( 45 ~; 4 |; 392 &)
% ( 16 <=>; 298 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 7 avg)
% Maximal term depth : 7 ( 1 avg)
% Number of predicates : 29 ( 27 usr; 1 prp; 0-3 aty)
% Number of functors : 75 ( 75 usr; 19 con; 0-4 aty)
% Number of variables : 365 ( 352 !; 13 ?)
% 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_7,axiom,
! [A,B] :
( ~ v1_xboole_0(k3_cqc_lang(A,B))
& v1_relat_1(k3_cqc_lang(A,B))
& v1_funct_1(k3_cqc_lang(A,B)) ) ).
fof(fc2_funct_7,axiom,
! [A,B,C,D] :
( ~ v1_xboole_0(k4_funct_4(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(rc1_funct_7,axiom,
? [A] :
( ~ v1_xboole_0(A)
& v1_relat_1(A)
& v2_relat_1(A)
& v1_funct_1(A)
& v1_finset_1(A)
& v1_finseq_1(A)
& v1_funcop_1(A) ) ).
fof(fc3_funct_7,axiom,
( v1_xboole_0(k1_xboole_0)
& v1_relat_1(k1_xboole_0)
& v3_relat_1(k1_xboole_0)
& v1_funct_1(k1_xboole_0)
& v2_funct_1(k1_xboole_0)
& v1_finset_1(k1_xboole_0)
& v1_finseq_1(k1_xboole_0)
& v1_funcop_1(k1_xboole_0) ) ).
fof(fc4_funct_7,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( ~ v1_xboole_0(k5_finseq_1(A))
& v1_relat_1(k5_finseq_1(A))
& v1_funct_1(k5_finseq_1(A))
& v1_finset_1(k5_finseq_1(A))
& v1_finseq_1(k5_finseq_1(A))
& v1_funcop_1(k5_finseq_1(A)) ) ) ).
fof(fc5_funct_7,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_relat_1(B)
& v1_funct_1(B) )
=> ( v1_relat_1(k10_finseq_1(A,B))
& v1_funct_1(k10_finseq_1(A,B))
& v1_finset_1(k10_finseq_1(A,B))
& v1_finseq_1(k10_finseq_1(A,B))
& v1_funcop_1(k10_finseq_1(A,B)) ) ) ).
fof(fc6_funct_7,axiom,
! [A,B,C] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_relat_1(B)
& v1_funct_1(B)
& v1_relat_1(C)
& v1_funct_1(C) )
=> ( v1_relat_1(k11_finseq_1(A,B,C))
& v1_funct_1(k11_finseq_1(A,B,C))
& v1_finset_1(k11_finseq_1(A,B,C))
& v1_finseq_1(k11_finseq_1(A,B,C))
& v1_funcop_1(k11_finseq_1(A,B,C)) ) ) ).
fof(fc7_funct_7,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& v1_relat_1(B)
& v1_funct_1(B) )
=> ( v1_relat_1(k2_finseq_2(A,B))
& v1_funct_1(k2_finseq_2(A,B))
& v1_finset_1(k2_finseq_2(A,B))
& v1_finseq_1(k2_finseq_2(A,B))
& v1_funcop_1(k2_finseq_2(A,B)) ) ) ).
fof(fc8_funct_7,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A)
& v1_funcop_1(A)
& v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B)
& v1_funcop_1(B) )
=> ( v1_relat_1(k7_finseq_1(A,B))
& v1_funct_1(k7_finseq_1(A,B))
& v1_finset_1(k7_finseq_1(A,B))
& v1_finseq_1(k7_finseq_1(A,B))
& v1_funcop_1(k7_finseq_1(A,B)) ) ) ).
fof(fc9_funct_7,axiom,
! [A,B,C] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A)
& m1_subset_1(B,k5_numbers) )
=> ( v1_relat_1(k2_funct_7(A,B,C))
& v1_funct_1(k2_funct_7(A,B,C))
& v1_finset_1(k2_funct_7(A,B,C))
& v1_finseq_1(k2_funct_7(A,B,C)) ) ) ).
fof(cc1_funct_7,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A)
& v1_funct_7(A) )
=> ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finset_1(A)
& v1_finseq_1(A)
& v1_funcop_1(A) ) ) ).
fof(cc2_funct_7,axiom,
! [A] :
( ( v1_xboole_0(A)
& v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finset_1(A)
& v1_finseq_1(A)
& v1_funcop_1(A)
& v1_funct_7(A) ) ) ).
fof(fc10_funct_7,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( ~ v1_xboole_0(k5_finseq_1(A))
& v1_relat_1(k5_finseq_1(A))
& v1_funct_1(k5_finseq_1(A))
& v1_finset_1(k5_finseq_1(A))
& v1_finseq_1(k5_finseq_1(A))
& v1_funcop_1(k5_finseq_1(A))
& v1_funct_7(k5_finseq_1(A)) ) ) ).
fof(rc2_funct_7,axiom,
? [A] :
( ~ v1_xboole_0(A)
& v1_relat_1(A)
& v2_relat_1(A)
& v1_funct_1(A)
& v1_finset_1(A)
& v1_finseq_1(A)
& v1_funcop_1(A)
& v1_funct_7(A) ) ).
fof(cc3_funct_7,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_relat_1(A)
& v2_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( m2_funct_7(B,A)
=> ( v2_relat_1(B)
& v1_funct_7(B) ) ) ) ).
fof(t1_funct_7,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( r1_tarski(k2_relat_1(A),B)
=> k5_relat_1(A,k6_partfun1(B)) = A ) ) ).
fof(t2_funct_7,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,B)
& m2_relset_1(C,A,B) )
=> ( v2_funct_1(C)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(B))
=> ( r1_tarski(E,k2_funct_2(A,B,C,D))
=> r1_tarski(k3_funct_2(A,B,C,E),D) ) ) ) ) ) ) ).
fof(t3_funct_7,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,B)
& m2_relset_1(C,A,B) )
=> ( v2_funct_1(C)
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(A))
=> ( r2_hidden(k8_funct_2(A,B,C,D),k2_funct_2(A,B,C,E))
=> r2_hidden(D,E) ) ) ) ) ) ) ) ).
fof(t4_funct_7,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,B)
& m2_relset_1(C,A,B) )
=> ( v2_funct_1(C)
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(A))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(B))
=> ( r2_hidden(k8_funct_2(A,B,C,D),k6_subset_1(B,k2_funct_2(A,B,C,E),F))
=> r2_hidden(D,k6_subset_1(A,E,k3_funct_2(A,B,C,F))) ) ) ) ) ) ) ) ) ).
fof(t5_funct_7,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,B)
& m2_relset_1(C,A,B) )
=> ( v2_funct_1(C)
=> ! [D] :
( m1_subset_1(D,B)
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(A))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(B))
=> ( r2_hidden(D,k6_subset_1(B,k2_funct_2(A,B,C,E),F))
=> r2_hidden(k1_funct_1(k2_funct_1(C),D),k6_subset_1(A,E,k3_funct_2(A,B,C,F))) ) ) ) ) ) ) ) ) ).
fof(t6_funct_7,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( r2_hidden(B,k1_relat_1(A))
=> k7_relat_1(A,k1_tarski(B)) = k3_cqc_lang(B,k1_funct_1(A,B)) ) ) ).
fof(t7_funct_7,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( r2_hidden(C,A)
=> k3_cqc_lang(C,k1_funct_1(B,C)) = k7_relat_1(B,k1_tarski(C)) ) ) ).
fof(t8_funct_7,axiom,
! [A,B,C] :
( m1_pboole(C,k2_zfmisc_1(A,B))
=> ! [D,E] :
( ( r2_hidden(D,A)
& r2_hidden(E,B) )
=> k9_funct_2(D,E,k1_binop_1(C,D,E)) = k7_relat_1(C,k2_zfmisc_1(k1_tarski(D),k1_tarski(E))) ) ) ).
fof(t9_funct_7,axiom,
$true ).
fof(t10_funct_7,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_tarski(k2_relat_1(B),k1_relat_1(A))
& r1_tarski(k2_relat_1(C),k1_relat_1(A)) )
=> k5_relat_1(k1_funct_4(B,C),A) = k1_funct_4(k5_relat_1(B,A),k5_relat_1(C,A)) ) ) ) ) ).
fof(t11_funct_7,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) )
=> k5_relat_1(A,k1_funct_4(B,C)) = k1_funct_4(k5_relat_1(A,B),k5_relat_1(A,C)) ) ) ) ).
fof(t12_funct_7,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_xboole_0(k2_relat_1(A),k1_relat_1(B))
=> k5_relat_1(A,k1_funct_4(C,B)) = k5_relat_1(A,C) ) ) ) ) ).
fof(t13_funct_7,axiom,
! [A,B,C] :
( ~ r1_xboole_0(A,k2_relat_1(k1_funct_4(k6_partfun1(B),k2_funcop_1(A,C))))
=> r2_hidden(C,A) ) ).
fof(t14_funct_7,axiom,
! [A,B,C] :
~ ( A != B
& r2_hidden(A,k2_relat_1(k1_funct_4(k6_partfun1(C),k3_cqc_lang(A,B)))) ) ).
fof(t15_funct_7,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( k1_relat_1(C) = k2_xboole_0(A,k1_tarski(B))
=> C = k1_funct_4(k7_relat_1(C,A),k3_cqc_lang(B,k1_funct_1(C,B))) ) ) ).
fof(t16_funct_7,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B,C,D] : k1_funct_4(k1_funct_4(A,k2_funcop_1(B,C)),k2_funcop_1(B,D)) = k1_funct_4(A,k2_funcop_1(B,D)) ) ).
fof(t17_funct_7,axiom,
$true ).
fof(t18_funct_7,axiom,
k4_numbers != k3_finseq_2(k4_numbers) ).
fof(t19_funct_7,axiom,
k3_finseq_2(k1_xboole_0) = k1_tarski(k1_xboole_0) ).
fof(t20_funct_7,axiom,
! [A,B] :
( r2_hidden(k9_finseq_1(A),k3_finseq_2(B))
<=> r2_hidden(A,B) ) ).
fof(t21_funct_7,axiom,
! [A,B] :
( r1_tarski(k3_finseq_2(A),k3_finseq_2(B))
=> r1_tarski(A,B) ) ).
fof(t22_funct_7,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k5_numbers))
=> ( ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(B,A)
=> ( r1_xreal_0(B,C)
| r2_hidden(C,A) ) ) ) )
=> v1_card_1(A) ) ) ).
fof(t23_funct_7,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ? [C] :
( m2_subset_1(C,k1_zfmisc_1(A),B)
& ! [D] :
( m2_subset_1(D,k1_zfmisc_1(A),B)
=> ( r1_tarski(D,C)
=> D = C ) ) ) ) ) ).
fof(t24_funct_7,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ( k3_finseq_1(A) = k1_nat_1(k3_finseq_1(B),np__1)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(C,k4_finseq_1(B))
<=> ( r2_hidden(C,k4_finseq_1(A))
& r2_hidden(k1_nat_1(C,np__1),k4_finseq_1(A)) ) ) ) ) ) ) ).
fof(t25_funct_7,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ( v1_funcop_1(k7_finseq_1(A,B))
=> ( v1_funcop_1(A)
& v1_funcop_1(B) ) ) ) ) ).
fof(d1_funct_7,axiom,
! [A,B] :
( r2_hidden(A,B)
=> k1_funct_7(A,B) = A ) ).
fof(t26_funct_7,axiom,
! [A,B,C] :
( r2_hidden(A,k3_xboole_0(B,C))
=> k1_funct_7(A,B) = k1_funct_7(A,C) ) ).
fof(d2_funct_7,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( r1_funct_7(A,B,C)
<=> k7_relat_1(A,k4_xboole_0(k1_relat_1(A),C)) = k7_relat_1(B,k4_xboole_0(k1_relat_1(B),C)) ) ) ) ).
fof(t27_funct_7,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] : r1_funct_7(A,A,B) ) ).
fof(t28_funct_7,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( r1_funct_7(A,B,C)
=> r1_funct_7(B,A,C) ) ) ) ).
fof(t29_funct_7,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] :
( ( r1_funct_7(A,B,D)
& r1_funct_7(B,C,D) )
=> r1_funct_7(A,C,D) ) ) ) ) ).
fof(t30_funct_7,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( r1_funct_7(A,B,C)
=> k4_xboole_0(k1_relat_1(A),C) = k4_xboole_0(k1_relat_1(B),C) ) ) ) ).
fof(t31_funct_7,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(B),C)
=> r1_funct_7(A,k1_funct_4(A,B),C) ) ) ) ).
fof(d3_funct_7,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B,C] :
( ( r2_hidden(B,k1_relat_1(A))
=> k2_funct_7(A,B,C) = k1_funct_4(A,k3_cqc_lang(B,C)) )
& ( ~ r2_hidden(B,k1_relat_1(A))
=> k2_funct_7(A,B,C) = A ) ) ) ).
fof(t32_funct_7,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B,C] : k1_relat_1(k2_funct_7(A,C,B)) = k1_relat_1(A) ) ).
fof(t33_funct_7,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B,C] :
( r2_hidden(C,k1_relat_1(A))
=> k1_funct_1(k2_funct_7(A,C,B),C) = B ) ) ).
fof(t34_funct_7,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B,C,D] :
( C != D
=> k1_funct_1(k2_funct_7(A,C,B),D) = k1_funct_1(A,D) ) ) ).
fof(t35_funct_7,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B,C,D,E] :
( D != E
=> k2_funct_7(k2_funct_7(A,D,B),E,C) = k2_funct_7(k2_funct_7(A,E,C),D,B) ) ) ).
fof(t36_funct_7,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B,C,D] : k2_funct_7(k2_funct_7(A,D,B),D,C) = k2_funct_7(A,D,C) ) ).
fof(t37_funct_7,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] : k2_funct_7(A,B,k1_funct_1(A,B)) = A ) ).
fof(t38_funct_7,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k4_finseq_1(B))
=> k4_finseq_4(k5_numbers,A,k3_funct_7(A,B,D,C),D) = C ) ) ) ) ) ).
fof(t39_funct_7,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r2_hidden(E,k4_finseq_1(B))
=> ( D = E
| k4_finseq_4(k5_numbers,A,k3_funct_7(A,B,D,C),E) = k4_finseq_4(k5_numbers,A,B,E) ) ) ) ) ) ) ) ).
fof(t40_funct_7,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> k3_funct_7(A,B,E,k4_finseq_4(k5_numbers,A,B,E)) = B ) ) ) ) ) ).
fof(d4_funct_7,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B)
& v1_funcop_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( C = k4_funct_7(A,B)
<=> ? [D] :
( v1_funcop_1(D)
& m1_pboole(D,k5_numbers)
& C = k1_funct_1(D,k3_finseq_1(B))
& k1_funct_1(D,np__0) = k6_partfun1(A)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r2_hidden(k1_nat_1(E,np__1),k4_finseq_1(B))
=> ! [F] :
( ( v1_relat_1(F)
& v1_funct_1(F) )
=> ! [G] :
( ( v1_relat_1(G)
& v1_funct_1(G) )
=> ( ( F = k1_funct_1(D,E)
& G = k1_funct_1(B,k1_nat_1(E,np__1)) )
=> k1_funct_1(D,k1_nat_1(E,np__1)) = k5_relat_1(F,G) ) ) ) ) ) ) ) ) ) ).
fof(d5_funct_7,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A)
& v1_funcop_1(A) )
=> ! [B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ( C = k5_funct_7(A,B)
<=> ( k3_finseq_1(C) = k1_nat_1(k3_finseq_1(A),np__1)
& k1_funct_1(C,np__1) = B
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( ( v1_relat_1(E)
& v1_funct_1(E) )
=> ( ( r2_hidden(D,k4_finseq_1(A))
& E = k1_funct_1(A,D) )
=> k1_funct_1(C,k1_nat_1(D,np__1)) = k1_funct_1(E,k1_funct_1(C,D)) ) ) ) ) ) ) ) ).
fof(t41_funct_7,axiom,
! [A] : k4_funct_7(A,k1_xboole_0) = k6_partfun1(A) ).
fof(t42_funct_7,axiom,
! [A] : k5_funct_7(k1_xboole_0,A) = k9_finseq_1(A) ).
fof(t43_funct_7,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B)
& v1_funcop_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> k4_funct_7(A,k7_finseq_1(B,k9_finseq_1(C))) = k5_relat_1(k4_funct_7(A,B),C) ) ) ).
fof(t44_funct_7,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B)
& v1_funcop_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> k5_funct_7(k7_finseq_1(B,k9_finseq_1(C)),A) = k7_finseq_1(k5_funct_7(B,A),k9_finseq_1(k1_funct_1(C,k1_funct_1(k5_funct_7(B,A),k1_nat_1(k3_finseq_1(B),np__1))))) ) ) ).
fof(t45_funct_7,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B)
& v1_funcop_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> k4_funct_7(A,k7_finseq_1(k9_finseq_1(C),B)) = k5_relat_1(k7_relat_1(C,A),k4_funct_7(k9_relat_1(C,A),B)) ) ) ).
fof(t46_funct_7,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B)
& v1_funcop_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> k5_funct_7(k7_finseq_1(k9_finseq_1(C),B),A) = k7_finseq_1(k9_finseq_1(A),k5_funct_7(B,k1_funct_1(C,A))) ) ) ).
fof(t47_funct_7,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> k4_funct_7(A,k9_finseq_1(B)) = k5_relat_1(k6_partfun1(A),B) ) ).
fof(t48_funct_7,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r1_tarski(k1_relat_1(B),A)
=> k4_funct_7(A,k9_finseq_1(B)) = B ) ) ).
fof(t49_funct_7,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> k5_funct_7(k9_finseq_1(B),A) = k10_finseq_1(A,k1_funct_1(B,A)) ) ).
fof(t50_funct_7,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C)
& v1_funcop_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D)
& v1_finseq_1(D)
& v1_funcop_1(D) )
=> ( r1_tarski(k2_relat_1(k4_funct_7(A,C)),B)
=> k4_funct_7(A,k7_finseq_1(C,D)) = k5_relat_1(k4_funct_7(A,C),k4_funct_7(B,D)) ) ) ) ).
fof(t51_funct_7,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B)
& v1_funcop_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C)
& v1_funcop_1(C) )
=> k1_funct_1(k5_funct_7(k7_finseq_1(B,C),A),k1_nat_1(k3_finseq_1(k7_finseq_1(B,C)),np__1)) = k1_funct_1(k5_funct_7(C,k1_funct_1(k5_funct_7(B,A),k1_nat_1(k3_finseq_1(B),np__1))),k1_nat_1(k3_finseq_1(C),np__1)) ) ) ).
fof(t52_funct_7,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B)
& v1_funcop_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C)
& v1_funcop_1(C) )
=> k5_funct_7(k7_finseq_1(B,C),A) = k1_rewrite1(k5_funct_7(B,A),k5_funct_7(C,k1_funct_1(k5_funct_7(B,A),k1_nat_1(k3_finseq_1(B),np__1)))) ) ) ).
fof(t53_funct_7,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> k4_funct_7(A,k10_finseq_1(B,C)) = k5_relat_1(k6_partfun1(A),k5_relat_1(B,C)) ) ) ).
fof(t54_funct_7,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( ( r1_tarski(k1_relat_1(B),A)
| r1_tarski(k1_relat_1(k5_relat_1(B,C)),A) )
=> k4_funct_7(A,k10_finseq_1(B,C)) = k5_relat_1(B,C) ) ) ) ).
fof(t55_funct_7,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> k5_funct_7(k10_finseq_1(B,C),A) = k11_finseq_1(A,k1_funct_1(B,A),k1_funct_1(C,k1_funct_1(B,A))) ) ) ).
fof(t56_funct_7,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) )
=> k4_funct_7(A,k11_finseq_1(B,C,D)) = k5_relat_1(k6_partfun1(A),k5_relat_1(B,k5_relat_1(C,D))) ) ) ) ).
fof(t57_funct_7,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) )
=> ( ~ ( ~ r1_tarski(k1_relat_1(B),A)
& ~ r1_tarski(k1_relat_1(k5_relat_1(B,C)),A)
& ~ r1_tarski(k1_relat_1(k5_relat_1(B,k5_relat_1(C,D))),A) )
=> k4_funct_7(A,k11_finseq_1(B,C,D)) = k5_relat_1(B,k5_relat_1(C,D)) ) ) ) ) ).
fof(t58_funct_7,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) )
=> k5_funct_7(k11_finseq_1(B,C,D),A) = k7_finseq_1(k9_finseq_1(A),k11_finseq_1(k1_funct_1(B,A),k1_funct_1(C,k1_funct_1(B,A)),k1_funct_1(D,k1_funct_1(C,k1_funct_1(B,A))))) ) ) ) ).
fof(d6_funct_7,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( v1_xboole_0(A)
=> ( B = k6_funct_7(A)
<=> v1_xboole_0(B) ) )
& ( ~ v1_xboole_0(A)
=> ( B = k6_funct_7(A)
<=> B = k1_funct_5(k1_funct_1(A,np__1)) ) ) ) ) ).
fof(d7_funct_7,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( v1_xboole_0(A)
=> ( B = k7_funct_7(A)
<=> v1_xboole_0(B) ) )
& ( ~ v1_xboole_0(A)
=> ( B = k7_funct_7(A)
<=> B = k2_funct_5(k1_funct_1(A,k3_finseq_1(A))) ) ) ) ) ).
fof(t59_funct_7,axiom,
( k6_funct_7(k1_xboole_0) = k1_xboole_0
& k7_funct_7(k1_xboole_0) = k1_xboole_0 ) ).
fof(t60_funct_7,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ( k6_funct_7(k7_finseq_1(k9_finseq_1(A),B)) = k1_relat_1(A)
& k7_funct_7(k7_finseq_1(B,k9_finseq_1(A))) = k2_relat_1(A) ) ) ) ).
fof(t61_funct_7,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B)
& v1_funcop_1(B) )
=> ( B != k1_xboole_0
=> r1_tarski(k2_relat_1(k4_funct_7(A,B)),k7_funct_7(B)) ) ) ).
fof(d8_funct_7,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ( v1_funct_7(A)
<=> ? [B] :
( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B)
& k3_finseq_1(B) = k1_nat_1(k3_finseq_1(A),np__1)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(C,k4_finseq_1(A))
=> r2_hidden(k1_funct_1(A,C),k1_funct_2(k1_funct_1(B,C),k1_funct_1(B,k1_nat_1(C,np__1)))) ) ) ) ) ) ).
fof(t62_funct_7,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ( v1_funct_7(k7_finseq_1(A,B))
=> ( v1_funct_7(A)
& v1_funct_7(B) ) ) ) ) ).
fof(t63_funct_7,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B)
& v1_funct_7(B) )
=> ( B != k1_xboole_0
=> k1_relat_1(k4_funct_7(A,B)) = k3_xboole_0(k6_funct_7(B),A) ) ) ).
fof(t64_funct_7,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A)
& v1_funct_7(A) )
=> k1_relat_1(k4_funct_7(k6_funct_7(A),A)) = k6_funct_7(A) ) ).
fof(t65_funct_7,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A)
& v1_funct_7(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r1_tarski(k2_relat_1(B),k6_funct_7(A))
=> ( v1_relat_1(k7_finseq_1(k9_finseq_1(B),A))
& v1_funct_1(k7_finseq_1(k9_finseq_1(B),A))
& v1_finseq_1(k7_finseq_1(k9_finseq_1(B),A))
& v1_funct_7(k7_finseq_1(k9_finseq_1(B),A)) ) ) ) ) ).
fof(t66_funct_7,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A)
& v1_funct_7(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r1_tarski(k7_funct_7(A),k1_relat_1(B))
=> ( v1_relat_1(k7_finseq_1(A,k9_finseq_1(B)))
& v1_funct_1(k7_finseq_1(A,k9_finseq_1(B)))
& v1_finseq_1(k7_finseq_1(A,k9_finseq_1(B)))
& v1_funct_7(k7_finseq_1(A,k9_finseq_1(B))) ) ) ) ) ).
fof(t67_funct_7,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C)
& v1_funct_7(C) )
=> ( ( r2_hidden(A,k6_funct_7(C))
& r2_hidden(A,B) )
=> k1_funct_1(k5_funct_7(C,A),k1_nat_1(k3_finseq_1(C),np__1)) = k1_funct_1(k4_funct_7(B,C),A) ) ) ).
fof(d9_funct_7,axiom,
! [A,B] :
( ( v1_xboole_0(B)
=> v1_xboole_0(A) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C)
& v1_funct_7(C) )
=> ( m1_funct_7(C,A,B)
<=> ( k6_funct_7(C) = A
& r1_tarski(k7_funct_7(C),B) ) ) ) ) ).
fof(d10_funct_7,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_relat_1(A)
& v2_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ( m2_funct_7(B,A)
<=> ( k1_nat_1(k3_finseq_1(B),np__1) = k3_finseq_1(A)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(C,k4_finseq_1(B))
=> r2_hidden(k1_funct_1(B,C),k1_funct_2(k1_funct_1(A,C),k1_funct_1(A,k1_nat_1(C,np__1)))) ) ) ) ) ) ) ).
fof(t68_funct_7,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_relat_1(A)
& v2_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( m2_funct_7(B,A)
=> ( B != k1_xboole_0
=> ( k6_funct_7(B) = k1_funct_1(A,np__1)
& r1_tarski(k7_funct_7(B),k1_funct_1(A,k3_finseq_1(A))) ) ) ) ) ).
fof(t69_funct_7,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_relat_1(A)
& v2_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( m2_funct_7(B,A)
=> ( k1_relat_1(k4_funct_7(k1_funct_1(A,np__1),B)) = k1_funct_1(A,np__1)
& r1_tarski(k2_relat_1(k4_funct_7(k1_funct_1(A,np__1),B)),k1_funct_1(A,k3_finseq_1(A))) ) ) ) ).
fof(d11_funct_7,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( C = k9_funct_7(A,B)
<=> ? [D] :
( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k4_partfun1(k2_xboole_0(k1_relat_1(A),k2_relat_1(A)),k2_xboole_0(k1_relat_1(A),k2_relat_1(A))))
& m2_relset_1(D,k5_numbers,k4_partfun1(k2_xboole_0(k1_relat_1(A),k2_relat_1(A)),k2_xboole_0(k1_relat_1(A),k2_relat_1(A))))
& C = k8_funct_2(k5_numbers,k4_partfun1(k2_xboole_0(k1_relat_1(A),k2_relat_1(A)),k2_xboole_0(k1_relat_1(A),k2_relat_1(A))),D,B)
& k8_funct_2(k5_numbers,k4_partfun1(k2_xboole_0(k1_relat_1(A),k2_relat_1(A)),k2_xboole_0(k1_relat_1(A),k2_relat_1(A))),D,np__0) = k6_partfun1(k2_xboole_0(k1_relat_1(A),k2_relat_1(A)))
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ? [F] :
( v1_relat_1(F)
& v1_funct_1(F)
& F = k8_funct_2(k5_numbers,k4_partfun1(k2_xboole_0(k1_relat_1(A),k2_relat_1(A)),k2_xboole_0(k1_relat_1(A),k2_relat_1(A))),D,E)
& k8_funct_2(k5_numbers,k4_partfun1(k2_xboole_0(k1_relat_1(A),k2_relat_1(A)),k2_xboole_0(k1_relat_1(A),k2_relat_1(A))),D,k1_nat_1(E,np__1)) = k5_relat_1(A,F) ) ) ) ) ) ) ) ).
fof(t70_funct_7,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> k9_funct_7(A,np__0) = k6_partfun1(k2_xboole_0(k1_relat_1(A),k2_relat_1(A))) ) ).
fof(t71_funct_7,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k9_funct_7(A,k1_nat_1(B,np__1)) = k5_relat_1(A,k9_funct_7(A,B)) ) ) ).
fof(t72_funct_7,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> k9_funct_7(A,np__1) = A ) ).
fof(t73_funct_7,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k9_funct_7(A,k1_nat_1(B,np__1)) = k5_relat_1(k9_funct_7(A,B),A) ) ) ).
fof(t74_funct_7,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_tarski(k1_relat_1(k9_funct_7(A,B)),k2_xboole_0(k1_relat_1(A),k2_relat_1(A)))
& r1_tarski(k2_relat_1(k9_funct_7(A,B)),k2_xboole_0(k1_relat_1(A),k2_relat_1(A))) ) ) ) ).
fof(t75_funct_7,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( B != np__0
=> ( r1_tarski(k1_relat_1(k9_funct_7(A,B)),k1_relat_1(A))
& r1_tarski(k2_relat_1(k9_funct_7(A,B)),k2_relat_1(A)) ) ) ) ) ).
fof(t76_funct_7,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_tarski(k2_relat_1(A),k1_relat_1(A))
=> ( k1_relat_1(k9_funct_7(A,B)) = k1_relat_1(A)
& r1_tarski(k2_relat_1(k9_funct_7(A,B)),k1_relat_1(A)) ) ) ) ) ).
fof(t77_funct_7,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k5_relat_1(k6_partfun1(k2_xboole_0(k1_relat_1(A),k2_relat_1(A))),k9_funct_7(A,B)) = k9_funct_7(A,B) ) ) ).
fof(t78_funct_7,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k5_relat_1(k9_funct_7(A,B),k6_partfun1(k2_xboole_0(k1_relat_1(A),k2_relat_1(A)))) = k9_funct_7(A,B) ) ) ).
fof(t79_funct_7,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k5_relat_1(k9_funct_7(A,C),k9_funct_7(A,B)) = k9_funct_7(A,k1_nat_1(B,C)) ) ) ) ).
fof(t80_funct_7,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( B != np__0
=> k9_funct_7(k9_funct_7(A,C),B) = k9_funct_7(A,k2_nat_1(C,B)) ) ) ) ) ).
fof(t81_funct_7,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_tarski(k2_relat_1(A),k1_relat_1(A))
=> k9_funct_7(k9_funct_7(A,B),C) = k9_funct_7(A,k2_nat_1(B,C)) ) ) ) ) ).
fof(t82_funct_7,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k9_funct_7(k1_xboole_0,A) = k1_xboole_0 ) ).
fof(t83_funct_7,axiom,
! [A,B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k9_funct_7(k6_partfun1(A),B) = k6_partfun1(A) ) ).
fof(t84_funct_7,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( r1_xboole_0(k2_relat_1(A),k1_relat_1(A))
=> k9_funct_7(A,np__2) = k1_xboole_0 ) ) ).
fof(t85_funct_7,axiom,
! [A,B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,A)
& m2_relset_1(C,A,A) )
=> ( v1_funct_1(k9_funct_7(C,B))
& v1_funct_2(k9_funct_7(C,B),A,A)
& m2_relset_1(k9_funct_7(C,B),A,A) ) ) ) ).
fof(t86_funct_7,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,A)
& m2_relset_1(B,A,A) )
=> k9_funct_7(B,np__0) = k6_partfun1(A) ) ).
fof(t87_funct_7,axiom,
! [A,B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,A)
& m2_relset_1(D,A,A) )
=> k9_funct_7(k9_funct_7(D,B),C) = k9_funct_7(D,k2_nat_1(B,C)) ) ) ) ).
fof(t88_funct_7,axiom,
! [A,B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,A) )
=> ( v1_funct_1(k9_funct_7(C,B))
& m2_relset_1(k9_funct_7(C,B),A,A) ) ) ) ).
fof(t89_funct_7,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( r2_hidden(A,B)
& C = k2_funcop_1(B,A) )
=> ( D = np__0
| k9_funct_7(C,D) = C ) ) ) ) ).
fof(t90_funct_7,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k9_funct_7(A,B) = k4_funct_7(k2_xboole_0(k1_relat_1(A),k2_relat_1(A)),k2_finseq_2(B,A)) ) ) ).
fof(s1_funct_7,axiom,
( ! [A] :
( m1_subset_1(A,f1_s1_funct_7)
=> ! [B] :
( m1_subset_1(B,f2_s1_funct_7)
=> r2_hidden(f4_s1_funct_7(A,B),f3_s1_funct_7) ) )
=> ? [A] :
( v1_funct_1(A)
& v1_funct_2(A,k2_zfmisc_1(f1_s1_funct_7,f2_s1_funct_7),f3_s1_funct_7)
& m2_relset_1(A,k2_zfmisc_1(f1_s1_funct_7,f2_s1_funct_7),f3_s1_funct_7)
& ! [B] :
( m1_subset_1(B,f1_s1_funct_7)
=> ! [C] :
( m1_subset_1(C,f2_s1_funct_7)
=> k8_funct_2(k2_zfmisc_1(f1_s1_funct_7,f2_s1_funct_7),f3_s1_funct_7,A,k1_domain_1(f1_s1_funct_7,f2_s1_funct_7,B,C)) = f4_s1_funct_7(B,C) ) ) ) ) ).
fof(s5_funct_7,axiom,
( ( p1_s5_funct_7(k1_xboole_0)
& ! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A)
& v1_funcop_1(A) )
=> ( p1_s5_funct_7(A)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> p1_s5_funct_7(k7_finseq_1(A,k9_finseq_1(B))) ) ) ) )
=> ! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A)
& v1_funcop_1(A) )
=> p1_s5_funct_7(A) ) ) ).
fof(dt_m1_funct_7,axiom,
! [A,B,C] :
( m1_funct_7(C,A,B)
=> ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C)
& v1_funct_7(C) ) ) ).
fof(existence_m1_funct_7,axiom,
! [A,B] :
? [C] : m1_funct_7(C,A,B) ).
fof(dt_m2_funct_7,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_relat_1(A)
& v2_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( m2_funct_7(B,A)
=> ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) ) ) ) ).
fof(existence_m2_funct_7,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_relat_1(A)
& v2_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ? [B] : m2_funct_7(B,A) ) ).
fof(dt_k1_funct_7,axiom,
! [A,B] : m1_subset_1(k1_funct_7(A,B),B) ).
fof(dt_k2_funct_7,axiom,
! [A,B,C] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v1_relat_1(k2_funct_7(A,B,C))
& v1_funct_1(k2_funct_7(A,B,C)) ) ) ).
fof(dt_k3_funct_7,axiom,
! [A,B,C,D] :
( ( m1_finseq_1(B,A)
& m1_subset_1(C,k5_numbers)
& m1_subset_1(D,A) )
=> m2_finseq_1(k3_funct_7(A,B,C,D),A) ) ).
fof(redefinition_k3_funct_7,axiom,
! [A,B,C,D] :
( ( m1_finseq_1(B,A)
& m1_subset_1(C,k5_numbers)
& m1_subset_1(D,A) )
=> k3_funct_7(A,B,C,D) = k2_funct_7(B,C,D) ) ).
fof(dt_k4_funct_7,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B)
& v1_funcop_1(B) )
=> ( v1_relat_1(k4_funct_7(A,B))
& v1_funct_1(k4_funct_7(A,B)) ) ) ).
fof(dt_k5_funct_7,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A)
& v1_funcop_1(A) )
=> ( v1_relat_1(k5_funct_7(A,B))
& v1_funct_1(k5_funct_7(A,B))
& v1_finseq_1(k5_funct_7(A,B)) ) ) ).
fof(dt_k6_funct_7,axiom,
$true ).
fof(dt_k7_funct_7,axiom,
$true ).
fof(dt_k8_funct_7,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_funct_7(C,B,A) )
=> ( v1_funct_1(k8_funct_7(A,B,C))
& v1_funct_2(k8_funct_7(A,B,C),B,A)
& m2_relset_1(k8_funct_7(A,B,C),B,A) ) ) ).
fof(redefinition_k8_funct_7,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_funct_7(C,B,A) )
=> k8_funct_7(A,B,C) = k4_funct_7(B,C) ) ).
fof(dt_k9_funct_7,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& m1_subset_1(B,k5_numbers) )
=> ( v1_relat_1(k9_funct_7(A,B))
& v1_funct_1(k9_funct_7(A,B)) ) ) ).
fof(s2_funct_7,axiom,
( ( v1_finset_1(f1_s2_funct_7)
& ! [A] :
( m1_subset_1(A,f2_s2_funct_7)
=> ! [B] :
( m1_subset_1(B,f2_s2_funct_7)
=> ( f4_s2_funct_7(A) = f4_s2_funct_7(B)
=> A = B ) ) ) )
=> v1_finset_1(a_0_0_funct_7) ) ).
fof(s3_funct_7,axiom,
( ( ! [A] :
~ ( r2_hidden(A,f1_s3_funct_7)
& ! [B] :
( m1_subset_1(B,f2_s3_funct_7)
=> A != f3_s3_funct_7(B) ) )
& ! [A] :
( m1_subset_1(A,f2_s3_funct_7)
=> ! [B] :
( m1_subset_1(B,f2_s3_funct_7)
=> ( f3_s3_funct_7(A) = f3_s3_funct_7(B)
=> A = B ) ) ) )
=> r2_tarski(f1_s3_funct_7,a_0_1_funct_7) ) ).
fof(s4_funct_7,axiom,
( ( r1_tarski(f1_s4_funct_7,f2_s4_funct_7)
& ! [A] :
( m1_subset_1(A,f2_s4_funct_7)
=> ! [B] :
( m1_subset_1(B,f2_s4_funct_7)
=> ( f3_s4_funct_7(A) = f3_s4_funct_7(B)
=> A = B ) ) ) )
=> r2_tarski(f1_s4_funct_7,a_0_2_funct_7) ) ).
fof(fraenkel_a_0_0_funct_7,axiom,
! [A] :
( r2_hidden(A,a_0_0_funct_7)
<=> ? [B] :
( m1_subset_1(B,f2_s2_funct_7)
& A = f3_s2_funct_7(B)
& r2_hidden(f4_s2_funct_7(B),f1_s2_funct_7) ) ) ).
fof(fraenkel_a_0_1_funct_7,axiom,
! [A] :
( r2_hidden(A,a_0_1_funct_7)
<=> ? [B] :
( m1_subset_1(B,f2_s3_funct_7)
& A = B
& r2_hidden(f3_s3_funct_7(B),f1_s3_funct_7) ) ) ).
fof(fraenkel_a_0_2_funct_7,axiom,
! [A] :
( r2_hidden(A,a_0_2_funct_7)
<=> ? [B] :
( m1_subset_1(B,f2_s4_funct_7)
& A = f3_s4_funct_7(B)
& r2_hidden(B,f1_s4_funct_7) ) ) ).
%------------------------------------------------------------------------------