SET007 Axioms: SET007+859.ax
%------------------------------------------------------------------------------
% File : SET007+859 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Substitution in First-Order Formulas: Elementary Properties
% 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 : substut1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 146 ( 10 unt; 0 def)
% Number of atoms : 733 ( 163 equ)
% Maximal formula atoms : 35 ( 5 avg)
% Number of connectives : 628 ( 41 ~; 0 |; 230 &)
% ( 27 <=>; 330 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 6 avg)
% Maximal term depth : 8 ( 1 avg)
% Number of predicates : 35 ( 33 usr; 1 prp; 0-3 aty)
% Number of functors : 111 ( 111 usr; 24 con; 0-4 aty)
% Number of variables : 307 ( 269 !; 38 ?)
% 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_substut1,axiom,
~ v1_xboole_0(k1_substut1) ).
fof(fc2_substut1,axiom,
( ~ v1_xboole_0(k1_substut1)
& v1_fraenkel(k1_substut1) ) ).
fof(rc1_substut1,axiom,
? [A] :
( m1_subset_1(A,k1_substut1)
& v1_relat_1(A)
& v1_funct_1(A)
& v1_finset_1(A) ) ).
fof(fc3_substut1,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> v1_finset_1(k9_substut1(A)) ) ).
fof(rc2_substut1,axiom,
? [A] :
( ~ v1_xboole_0(A)
& v1_substut1(A) ) ).
fof(fc4_substut1,axiom,
( ~ v1_xboole_0(k16_substut1)
& v1_substut1(k16_substut1) ) ).
fof(fc5_substut1,axiom,
! [A,B,C,D] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k7_qc_lang1(A))
& m1_qc_lang1(C,A)
& m1_subset_1(D,k1_substut1) )
=> v4_substut1(k17_substut1(B,C,D)) ) ).
fof(fc6_substut1,axiom,
! [A] :
( m1_subset_1(A,k16_substut1)
=> v5_substut1(k20_substut1(A)) ) ).
fof(fc7_substut1,axiom,
~ v1_xboole_0(k38_substut1) ).
fof(d1_substut1,axiom,
k1_substut1 = k4_partfun1(k2_qc_lang1,k2_qc_lang1) ).
fof(d2_substut1,axiom,
! [A] :
( m1_subset_1(A,k1_substut1)
=> k2_substut1(A) = A ) ).
fof(t1_substut1,axiom,
! [A,B] :
( m1_subset_1(B,k1_substut1)
=> ( r2_hidden(A,k1_relat_1(B))
=> r2_hidden(k1_funct_1(B,A),k2_qc_lang1) ) ) ).
fof(d3_substut1,axiom,
! [A] :
( m2_finseq_1(A,k1_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k1_substut1)
=> ! [C] :
( m2_finseq_1(C,k1_qc_lang1)
=> ( C = k3_substut1(A,B)
<=> ( k3_finseq_1(C) = k3_finseq_1(A)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,D)
& r1_xreal_0(D,k3_finseq_1(A)) )
=> ( ( r2_hidden(k1_funct_1(A,D),k1_relat_1(B))
=> k1_funct_1(C,D) = k1_funct_1(B,k1_funct_1(A,D)) )
& ( ~ r2_hidden(k1_funct_1(A,D),k1_relat_1(B))
=> k1_funct_1(C,D) = k1_funct_1(A,D) ) ) ) ) ) ) ) ) ) ).
fof(d4_substut1,axiom,
! [A] :
( m2_finseq_1(A,k2_qc_lang1)
=> k4_substut1(A) = A ) ).
fof(d5_substut1,axiom,
! [A] :
( m2_finseq_1(A,k2_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k1_substut1)
=> k5_substut1(A,B) = k3_substut1(k4_substut1(A),B) ) ) ).
fof(d8_substut1,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k2_qc_lang1))
=> ( B = k9_substut1(A)
<=> ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,k8_qc_lang1,k1_zfmisc_1(k2_qc_lang1))
& m2_relset_1(C,k8_qc_lang1,k1_zfmisc_1(k2_qc_lang1))
& B = k8_funct_2(k8_qc_lang1,k1_zfmisc_1(k2_qc_lang1),C,A)
& ! [D] :
( m1_subset_1(D,k8_qc_lang1)
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(k2_qc_lang1))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(k2_qc_lang1))
=> ( ( D = k9_cqc_lang
=> k8_funct_2(k8_qc_lang1,k1_zfmisc_1(k2_qc_lang1),C,D) = k1_subset_1(k2_qc_lang1) )
& ( v2_qc_lang1(D)
=> k8_funct_2(k8_qc_lang1,k1_zfmisc_1(k2_qc_lang1),C,D) = k8_substut1(k16_qc_lang1(D)) )
& ( ( v3_qc_lang1(D)
& E = k8_funct_2(k8_qc_lang1,k1_zfmisc_1(k2_qc_lang1),C,k17_qc_lang1(D)) )
=> k8_funct_2(k8_qc_lang1,k1_zfmisc_1(k2_qc_lang1),C,D) = E )
& ( ( v4_qc_lang1(D)
& E = k8_funct_2(k8_qc_lang1,k1_zfmisc_1(k2_qc_lang1),C,k18_qc_lang1(D))
& F = k8_funct_2(k8_qc_lang1,k1_zfmisc_1(k2_qc_lang1),C,k19_qc_lang1(D)) )
=> k8_funct_2(k8_qc_lang1,k1_zfmisc_1(k2_qc_lang1),C,D) = k4_subset_1(k2_qc_lang1,E,F) )
& ( ( v5_qc_lang1(D)
& E = k8_funct_2(k8_qc_lang1,k1_zfmisc_1(k2_qc_lang1),C,k21_qc_lang1(D)) )
=> k8_funct_2(k8_qc_lang1,k1_zfmisc_1(k2_qc_lang1),C,D) = k4_subset_1(k2_qc_lang1,E,k6_domain_1(k2_qc_lang1,k20_qc_lang1(D))) ) ) ) ) ) ) ) ) ) ).
fof(t2_substut1,axiom,
k9_substut1(k9_cqc_lang) = k1_xboole_0 ).
fof(t3_substut1,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ( v2_qc_lang1(A)
=> k9_substut1(A) = k8_substut1(k16_qc_lang1(A)) ) ) ).
fof(t4_substut1,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ( v3_qc_lang1(A)
=> k9_substut1(A) = k9_substut1(k17_qc_lang1(A)) ) ) ).
fof(t5_substut1,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ( v4_qc_lang1(A)
=> k9_substut1(A) = k4_subset_1(k2_qc_lang1,k9_substut1(k18_qc_lang1(A)),k9_substut1(k19_qc_lang1(A))) ) ) ).
fof(t6_substut1,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ( v5_qc_lang1(A)
=> k9_substut1(A) = k4_subset_1(k2_qc_lang1,k9_substut1(k21_qc_lang1(A)),k6_domain_1(k2_qc_lang1,k20_qc_lang1(A))) ) ) ).
fof(d11_substut1,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( ( v1_finset_1(B)
& m1_subset_1(B,k1_substut1) )
=> k12_substut1(A,B) = k4_xboole_0(k5_numbers,k4_subset_1(k5_numbers,k10_substut1(A),k11_substut1(B))) ) ) ).
fof(d12_substut1,axiom,
! [A] :
( ( v1_finset_1(A)
& m1_subset_1(A,k1_substut1) )
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> k13_substut1(A,B) = k10_cqc_sim1(k12_substut1(B,A)) ) ) ).
fof(d13_substut1,axiom,
! [A] :
( m2_subset_1(A,k1_qc_lang1,k2_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> ! [C] :
( ( v1_finset_1(C)
& m1_subset_1(C,k1_substut1) )
=> ( ? [D] :
( m1_subset_1(D,k1_substut1)
& C = k7_substut1(A,k14_qc_lang1(A,B),D) )
=> ( ( r2_hidden(A,k2_relat_1(C))
=> k14_substut1(A,B,C) = k2_xboole_0(C,k6_domain_1(k2_zfmisc_1(k2_qc_lang1,k2_qc_lang1),k1_domain_1(k2_qc_lang1,k2_qc_lang1,A,k2_qc_lang3(k13_substut1(C,B))))) )
& ( ~ r2_hidden(A,k2_relat_1(C))
=> k14_substut1(A,B,C) = k2_xboole_0(C,k6_domain_1(k2_zfmisc_1(k2_qc_lang1,k2_qc_lang1),k1_domain_1(k2_qc_lang1,k2_qc_lang1,A,A))) ) ) ) ) ) ) ).
fof(d14_substut1,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k1_substut1)
=> ! [C] :
( r1_substut1(A,B,C)
<=> ( ( v5_qc_lang1(A)
=> C = k14_substut1(k20_qc_lang1(A),k21_qc_lang1(A),k7_substut1(k20_qc_lang1(A),A,B)) )
& ( ~ v5_qc_lang1(A)
=> C = k1_xboole_0 ) ) ) ) ) ).
fof(d15_substut1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( A = k15_substut1
<=> ! [B] :
( r2_hidden(B,A)
<=> ? [C] :
( m1_subset_1(C,k8_qc_lang1)
& ? [D] :
( m1_subset_1(D,k1_substut1)
& ? [E] :
( B = k4_tarski(k1_domain_1(k8_qc_lang1,k1_substut1,C,D),E)
& r1_substut1(C,D,E) ) ) ) ) ) ) ).
fof(t7_substut1,axiom,
( m1_subset_1(k2_zfmisc_1(k8_qc_lang1,k1_substut1),k1_zfmisc_1(k2_zfmisc_1(k13_finseq_1(k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers)),k1_substut1)))
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k5_qc_lang1,k7_qc_lang1(A))
=> ! [C] :
( m1_qc_lang1(C,A)
=> ! [D] :
( m1_subset_1(D,k1_substut1)
=> r2_hidden(k4_tarski(k7_finseq_1(k12_finseq_1(k7_qc_lang1(A),B),C),D),k2_zfmisc_1(k8_qc_lang1,k1_substut1)) ) ) ) )
& ! [A] :
( m1_subset_1(A,k1_substut1)
=> r2_hidden(k1_domain_1(k1_zfmisc_1(k2_zfmisc_1(k5_numbers,k2_zfmisc_1(k5_numbers,k5_numbers))),k1_substut1,k12_finseq_1(k2_zfmisc_1(k5_numbers,k5_numbers),k1_domain_1(k5_numbers,k5_numbers,np__0,np__0)),A),k2_zfmisc_1(k8_qc_lang1,k1_substut1)) )
& ! [A] :
( m2_finseq_1(A,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))
=> ! [B] :
( m1_subset_1(B,k1_substut1)
=> ( r2_hidden(k1_domain_1(k1_zfmisc_1(k2_zfmisc_1(k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))),k1_substut1,A,B),k2_zfmisc_1(k8_qc_lang1,k1_substut1))
=> r2_hidden(k1_domain_1(k1_zfmisc_1(k2_zfmisc_1(k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))),k1_substut1,k8_finseq_1(k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k12_finseq_1(k2_zfmisc_1(k5_numbers,k5_numbers),k1_domain_1(k5_numbers,k5_numbers,np__1,np__0)),A),B),k2_zfmisc_1(k8_qc_lang1,k1_substut1)) ) ) )
& ! [A] :
( m2_finseq_1(A,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))
=> ! [B] :
( m2_finseq_1(B,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))
=> ! [C] :
( m1_subset_1(C,k1_substut1)
=> ( ( r2_hidden(k1_domain_1(k1_zfmisc_1(k2_zfmisc_1(k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))),k1_substut1,A,C),k2_zfmisc_1(k8_qc_lang1,k1_substut1))
& r2_hidden(k1_domain_1(k1_zfmisc_1(k2_zfmisc_1(k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))),k1_substut1,B,C),k2_zfmisc_1(k8_qc_lang1,k1_substut1)) )
=> r2_hidden(k1_domain_1(k1_zfmisc_1(k2_zfmisc_1(k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))),k1_substut1,k8_finseq_1(k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k8_finseq_1(k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k12_finseq_1(k2_zfmisc_1(k5_numbers,k5_numbers),k1_domain_1(k5_numbers,k5_numbers,np__2,np__0)),A),B),C),k2_zfmisc_1(k8_qc_lang1,k1_substut1)) ) ) ) )
& ! [A] :
( m2_subset_1(A,k1_qc_lang1,k2_qc_lang1)
=> ! [B] :
( m2_finseq_1(B,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))
=> ! [C] :
( m1_subset_1(C,k1_substut1)
=> ( r2_hidden(k4_tarski(B,k1_funct_1(k15_substut1,k4_tarski(k7_finseq_1(k7_finseq_1(k12_finseq_1(k2_zfmisc_1(k5_numbers,k5_numbers),k1_domain_1(k5_numbers,k5_numbers,np__3,np__0)),k12_finseq_1(k2_qc_lang1,A)),B),C))),k2_zfmisc_1(k8_qc_lang1,k1_substut1))
=> r2_hidden(k4_tarski(k7_finseq_1(k7_finseq_1(k12_finseq_1(k2_zfmisc_1(k5_numbers,k5_numbers),k1_domain_1(k5_numbers,k5_numbers,np__3,np__0)),k12_finseq_1(k2_qc_lang1,A)),B),C),k2_zfmisc_1(k8_qc_lang1,k1_substut1)) ) ) ) ) ) ).
fof(d16_substut1,axiom,
! [A] :
( v1_substut1(A)
<=> ( m1_subset_1(A,k1_zfmisc_1(k2_zfmisc_1(k13_finseq_1(k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers)),k1_substut1)))
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k5_qc_lang1,k7_qc_lang1(B))
=> ! [D] :
( m1_qc_lang1(D,B)
=> ! [E] :
( m1_subset_1(E,k1_substut1)
=> r2_hidden(k4_tarski(k7_finseq_1(k12_finseq_1(k7_qc_lang1(B),C),D),E),A) ) ) ) )
& ! [B] :
( m1_subset_1(B,k1_substut1)
=> r2_hidden(k1_domain_1(k1_zfmisc_1(k2_zfmisc_1(k5_numbers,k2_zfmisc_1(k5_numbers,k5_numbers))),k1_substut1,k12_finseq_1(k2_zfmisc_1(k5_numbers,k5_numbers),k1_domain_1(k5_numbers,k5_numbers,np__0,np__0)),B),A) )
& ! [B] :
( m2_finseq_1(B,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))
=> ! [C] :
( m1_subset_1(C,k1_substut1)
=> ( r2_hidden(k1_domain_1(k1_zfmisc_1(k2_zfmisc_1(k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))),k1_substut1,B,C),A)
=> r2_hidden(k1_domain_1(k1_zfmisc_1(k2_zfmisc_1(k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))),k1_substut1,k8_finseq_1(k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k12_finseq_1(k2_zfmisc_1(k5_numbers,k5_numbers),k1_domain_1(k5_numbers,k5_numbers,np__1,np__0)),B),C),A) ) ) )
& ! [B] :
( m2_finseq_1(B,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))
=> ! [C] :
( m2_finseq_1(C,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))
=> ! [D] :
( m1_subset_1(D,k1_substut1)
=> ( ( r2_hidden(k1_domain_1(k1_zfmisc_1(k2_zfmisc_1(k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))),k1_substut1,B,D),A)
& r2_hidden(k1_domain_1(k1_zfmisc_1(k2_zfmisc_1(k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))),k1_substut1,C,D),A) )
=> r2_hidden(k1_domain_1(k1_zfmisc_1(k2_zfmisc_1(k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))),k1_substut1,k8_finseq_1(k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k8_finseq_1(k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k12_finseq_1(k2_zfmisc_1(k5_numbers,k5_numbers),k1_domain_1(k5_numbers,k5_numbers,np__2,np__0)),B),C),D),A) ) ) ) )
& ! [B] :
( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1)
=> ! [C] :
( m2_finseq_1(C,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))
=> ! [D] :
( m1_subset_1(D,k1_substut1)
=> ( r2_hidden(k4_tarski(C,k1_funct_1(k15_substut1,k4_tarski(k7_finseq_1(k7_finseq_1(k12_finseq_1(k2_zfmisc_1(k5_numbers,k5_numbers),k1_domain_1(k5_numbers,k5_numbers,np__3,np__0)),k12_finseq_1(k2_qc_lang1,B)),C),D))),A)
=> r2_hidden(k4_tarski(k7_finseq_1(k7_finseq_1(k12_finseq_1(k2_zfmisc_1(k5_numbers,k5_numbers),k1_domain_1(k5_numbers,k5_numbers,np__3,np__0)),k12_finseq_1(k2_qc_lang1,B)),C),D),A) ) ) ) ) ) ) ).
fof(d17_substut1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( A = k16_substut1
<=> ( v1_substut1(A)
& ! [B] :
( ~ v1_xboole_0(B)
=> ( v1_substut1(B)
=> r1_tarski(A,B) ) ) ) ) ) ).
fof(t8_substut1,axiom,
! [A] :
( m1_subset_1(A,k16_substut1)
=> ? [B] :
( m1_subset_1(B,k8_qc_lang1)
& ? [C] :
( m1_subset_1(C,k1_substut1)
& A = k1_domain_1(k8_qc_lang1,k1_substut1,B,C) ) ) ) ).
fof(d18_substut1,axiom,
! [A] :
( m1_subset_1(A,k5_qc_lang1)
=> ! [B] :
( m2_finseq_1(B,k1_qc_lang1)
=> ! [C] :
( m1_subset_1(C,k1_substut1)
=> ( k6_qc_lang1(A) = k3_finseq_1(B)
=> k17_substut1(A,B,C) = k1_domain_1(k8_qc_lang1,k1_substut1,k9_qc_lang1(A,B),C) ) ) ) ) ).
fof(t9_substut1,axiom,
! [A] :
( m1_subset_1(A,k1_substut1)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k5_qc_lang1,k7_qc_lang1(B))
=> ! [D] :
( m1_qc_lang1(D,B)
=> k17_substut1(C,D,A) = k1_domain_1(k8_qc_lang1,k1_substut1,k9_qc_lang1(C,D),A) ) ) ) ) ).
fof(d19_substut1,axiom,
! [A] :
( m1_subset_1(A,k16_substut1)
=> ( v2_substut1(A)
<=> ? [B] :
( m1_subset_1(B,k1_substut1)
& A = k1_domain_1(k7_cqc_lang,k1_substut1,k9_cqc_lang,B) ) ) ) ).
fof(t10_substut1,axiom,
! [A] :
( m1_subset_1(A,k16_substut1)
=> A = k1_domain_1(k8_qc_lang1,k1_substut1,k18_substut1(A),k19_substut1(A)) ) ).
fof(d20_substut1,axiom,
! [A] :
( m1_subset_1(A,k16_substut1)
=> k20_substut1(A) = k1_domain_1(k8_qc_lang1,k1_substut1,k12_qc_lang1(k18_substut1(A)),k19_substut1(A)) ) ).
fof(d21_substut1,axiom,
! [A] :
( m1_subset_1(A,k16_substut1)
=> ! [B] :
( m1_subset_1(B,k16_substut1)
=> ( k19_substut1(A) = k19_substut1(B)
=> k21_substut1(A,B) = k1_domain_1(k8_qc_lang1,k1_substut1,k13_qc_lang1(k18_substut1(A),k18_substut1(B)),k19_substut1(A)) ) ) ) ).
fof(d22_substut1,axiom,
! [A] :
( m1_subset_1(A,k2_zfmisc_1(k16_substut1,k2_qc_lang1))
=> ( v3_substut1(A)
<=> ? [B] :
( m1_subset_1(B,k1_substut1)
& k19_substut1(k22_substut1(A)) = k1_funct_1(k15_substut1,k1_domain_1(k8_qc_lang1,k1_substut1,k14_qc_lang1(k23_substut1(A),k18_substut1(k22_substut1(A))),B)) ) ) ) ).
fof(d23_substut1,axiom,
! [A] :
( m1_subset_1(A,k2_zfmisc_1(k16_substut1,k2_qc_lang1))
=> ( v3_substut1(A)
=> ! [B] :
( m1_subset_1(B,k1_substut1)
=> ( m1_substut1(B,A)
<=> k19_substut1(k22_substut1(A)) = k1_funct_1(k15_substut1,k1_domain_1(k8_qc_lang1,k1_substut1,k14_qc_lang1(k23_substut1(A),k18_substut1(k22_substut1(A))),B)) ) ) ) ) ).
fof(d24_substut1,axiom,
! [A] :
( m1_subset_1(A,k2_zfmisc_1(k16_substut1,k2_qc_lang1))
=> ! [B] :
( m1_substut1(B,A)
=> ( v3_substut1(A)
=> k24_substut1(A,B) = k1_domain_1(k8_qc_lang1,k1_substut1,k14_qc_lang1(k23_substut1(A),k18_substut1(k22_substut1(A))),B) ) ) ) ).
fof(d25_substut1,axiom,
! [A] :
( m1_subset_1(A,k16_substut1)
=> ( v4_substut1(A)
<=> ? [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
& ? [C] :
( m2_subset_1(C,k5_qc_lang1,k7_qc_lang1(B))
& ? [D] :
( m1_qc_lang1(D,B)
& ? [E] :
( m1_subset_1(E,k1_substut1)
& A = k17_substut1(C,D,E) ) ) ) ) ) ) ).
fof(t11_substut1,axiom,
! [A] :
( m1_subset_1(A,k16_substut1)
=> ( v4_substut1(A)
=> v2_qc_lang1(k18_substut1(A)) ) ) ).
fof(d26_substut1,axiom,
! [A] :
( m1_subset_1(A,k16_substut1)
=> ( v5_substut1(A)
<=> ? [B] :
( m1_subset_1(B,k16_substut1)
& A = k20_substut1(B) ) ) ) ).
fof(d27_substut1,axiom,
! [A] :
( m1_subset_1(A,k16_substut1)
=> ( v6_substut1(A)
<=> ? [B] :
( m1_subset_1(B,k16_substut1)
& ? [C] :
( m1_subset_1(C,k16_substut1)
& A = k21_substut1(B,C)
& k19_substut1(B) = k19_substut1(C) ) ) ) ) ).
fof(d28_substut1,axiom,
! [A] :
( v7_substut1(A)
<=> ? [B] :
( m1_subset_1(B,k2_zfmisc_1(k16_substut1,k2_qc_lang1))
& ? [C] :
( m1_substut1(C,B)
& A = k24_substut1(B,C)
& v3_substut1(B) ) ) ) ).
fof(t12_substut1,axiom,
! [A] :
( m1_subset_1(A,k16_substut1)
=> ~ ( ~ v2_substut1(A)
& ~ v4_substut1(A)
& ~ v5_substut1(A)
& ~ v6_substut1(A)
& ~ v7_substut1(A) ) ) ).
fof(d29_substut1,axiom,
! [A] :
( m1_subset_1(A,k16_substut1)
=> ( v4_substut1(A)
=> ! [B] :
( m2_finseq_1(B,k1_qc_lang1)
=> ( B = k26_substut1(A)
<=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& ? [D] :
( m2_subset_1(D,k5_qc_lang1,k7_qc_lang1(C))
& ? [E] :
( m1_qc_lang1(E,C)
& ? [F] :
( m1_subset_1(F,k1_substut1)
& B = E
& A = k17_substut1(D,E,F) ) ) ) ) ) ) ) ) ).
fof(d30_substut1,axiom,
! [A] :
( m1_subset_1(A,k16_substut1)
=> ( v5_substut1(A)
=> ! [B] :
( m1_subset_1(B,k16_substut1)
=> ( B = k27_substut1(A)
<=> A = k20_substut1(B) ) ) ) ) ).
fof(d31_substut1,axiom,
! [A] :
( m1_subset_1(A,k16_substut1)
=> ( v6_substut1(A)
=> ! [B] :
( m1_subset_1(B,k16_substut1)
=> ( B = k28_substut1(A)
<=> ? [C] :
( m1_subset_1(C,k16_substut1)
& A = k21_substut1(B,C)
& k19_substut1(B) = k19_substut1(C) ) ) ) ) ) ).
fof(d32_substut1,axiom,
! [A] :
( m1_subset_1(A,k16_substut1)
=> ( v6_substut1(A)
=> ! [B] :
( m1_subset_1(B,k16_substut1)
=> ( B = k29_substut1(A)
<=> ? [C] :
( m1_subset_1(C,k16_substut1)
& A = k21_substut1(C,B)
& k19_substut1(C) = k19_substut1(B) ) ) ) ) ) ).
fof(d33_substut1,axiom,
! [A] :
( v7_substut1(A)
=> ! [B] :
( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1)
=> ( B = k30_substut1(A)
<=> ? [C] :
( m1_subset_1(C,k2_zfmisc_1(k16_substut1,k2_qc_lang1))
& ? [D] :
( m1_substut1(D,C)
& A = k24_substut1(C,D)
& k23_substut1(C) = B
& v3_substut1(C) ) ) ) ) ) ).
fof(d34_substut1,axiom,
! [A] :
( v7_substut1(A)
=> ! [B] :
( m1_subset_1(B,k16_substut1)
=> ( B = k31_substut1(A)
<=> ? [C] :
( m1_subset_1(C,k2_zfmisc_1(k16_substut1,k2_qc_lang1))
& ? [D] :
( m1_substut1(D,C)
& A = k24_substut1(C,D)
& k22_substut1(C) = B
& v3_substut1(C) ) ) ) ) ) ).
fof(t13_substut1,axiom,
! [A] :
( m1_subset_1(A,k16_substut1)
=> ! [B] :
( m1_subset_1(B,k16_substut1)
=> ( k19_substut1(A) = k19_substut1(B)
=> v6_substut1(k21_substut1(A,B)) ) ) ) ).
fof(t14_substut1,axiom,
! [A] :
( m1_subset_1(A,k2_zfmisc_1(k16_substut1,k2_qc_lang1))
=> ! [B] :
( m1_substut1(B,A)
=> ( v3_substut1(A)
=> v7_substut1(k24_substut1(A,B)) ) ) ) ).
fof(t15_substut1,axiom,
! [A] :
( m1_subset_1(A,k16_substut1)
=> ! [B] :
( m1_subset_1(B,k16_substut1)
=> ( k20_substut1(A) = k20_substut1(B)
=> A = B ) ) ) ).
fof(t16_substut1,axiom,
! [A] :
( m1_subset_1(A,k16_substut1)
=> k27_substut1(k20_substut1(A)) = A ) ).
fof(t17_substut1,axiom,
! [A] :
( m1_subset_1(A,k16_substut1)
=> ! [B] :
( m1_subset_1(B,k16_substut1)
=> ! [C] :
( m1_subset_1(C,k16_substut1)
=> ! [D] :
( m1_subset_1(D,k16_substut1)
=> ( ( k19_substut1(A) = k19_substut1(B)
& k19_substut1(C) = k19_substut1(D)
& k21_substut1(A,B) = k21_substut1(C,D) )
=> ( A = C
& B = D ) ) ) ) ) ) ).
fof(t18_substut1,axiom,
! [A] :
( m1_subset_1(A,k16_substut1)
=> ! [B] :
( m1_subset_1(B,k16_substut1)
=> ( k19_substut1(A) = k19_substut1(B)
=> k28_substut1(k21_substut1(A,B)) = A ) ) ) ).
fof(t19_substut1,axiom,
! [A] :
( m1_subset_1(A,k16_substut1)
=> ! [B] :
( m1_subset_1(B,k16_substut1)
=> ( k19_substut1(A) = k19_substut1(B)
=> k29_substut1(k21_substut1(A,B)) = B ) ) ) ).
fof(t20_substut1,axiom,
! [A] :
( m1_subset_1(A,k2_zfmisc_1(k16_substut1,k2_qc_lang1))
=> ! [B] :
( m1_subset_1(B,k2_zfmisc_1(k16_substut1,k2_qc_lang1))
=> ! [C] :
( m1_substut1(C,A)
=> ! [D] :
( m1_substut1(D,B)
=> ( ( v3_substut1(A)
& v3_substut1(B)
& k24_substut1(A,C) = k24_substut1(B,D) )
=> A = B ) ) ) ) ) ).
fof(t21_substut1,axiom,
! [A] :
( m1_subset_1(A,k2_zfmisc_1(k16_substut1,k2_qc_lang1))
=> ! [B] :
( m1_substut1(B,A)
=> ( v3_substut1(A)
=> k31_substut1(k24_substut1(A,B)) = k22_substut1(A) ) ) ) ).
fof(t22_substut1,axiom,
! [A] :
( m1_subset_1(A,k16_substut1)
=> ~ ( v5_substut1(A)
& r1_xreal_0(k3_finseq_1(k10_qc_lang1(k18_substut1(A))),k3_finseq_1(k10_qc_lang1(k18_substut1(k27_substut1(A))))) ) ) ).
fof(t23_substut1,axiom,
! [A] :
( m1_subset_1(A,k16_substut1)
=> ( v6_substut1(A)
=> ( ~ r1_xreal_0(k3_finseq_1(k10_qc_lang1(k18_substut1(A))),k3_finseq_1(k10_qc_lang1(k18_substut1(k28_substut1(A)))))
& ~ r1_xreal_0(k3_finseq_1(k10_qc_lang1(k18_substut1(A))),k3_finseq_1(k10_qc_lang1(k18_substut1(k29_substut1(A))))) ) ) ) ).
fof(t24_substut1,axiom,
! [A] :
( m1_subset_1(A,k16_substut1)
=> ~ ( v7_substut1(A)
& r1_xreal_0(k3_finseq_1(k10_qc_lang1(k18_substut1(A))),k3_finseq_1(k10_qc_lang1(k18_substut1(k31_substut1(A))))) ) ) ).
fof(t25_substut1,axiom,
! [A] :
( m1_subset_1(A,k16_substut1)
=> ( ( v2_substut1(A)
=> k1_mcart_1(k1_funct_1(k10_qc_lang1(k18_substut1(A)),np__1)) = np__0 )
& ~ ( v4_substut1(A)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ m2_subset_1(k1_funct_1(k10_qc_lang1(k18_substut1(A)),np__1),k5_qc_lang1,k7_qc_lang1(B)) ) )
& ( v5_substut1(A)
=> k1_mcart_1(k1_funct_1(k10_qc_lang1(k18_substut1(A)),np__1)) = np__1 )
& ( v6_substut1(A)
=> k1_mcart_1(k1_funct_1(k10_qc_lang1(k18_substut1(A)),np__1)) = np__2 )
& ( v7_substut1(A)
=> k1_mcart_1(k1_funct_1(k10_qc_lang1(k18_substut1(A)),np__1)) = np__3 ) ) ) ).
fof(t26_substut1,axiom,
! [A] :
( m1_subset_1(A,k16_substut1)
=> ( v4_substut1(A)
=> ( k1_mcart_1(k1_funct_1(k10_qc_lang1(k18_substut1(A)),np__1)) != np__0
& k1_mcart_1(k1_funct_1(k10_qc_lang1(k18_substut1(A)),np__1)) != np__1
& k1_mcart_1(k1_funct_1(k10_qc_lang1(k18_substut1(A)),np__1)) != np__2
& k1_mcart_1(k1_funct_1(k10_qc_lang1(k18_substut1(A)),np__1)) != np__3 ) ) ) ).
fof(t27_substut1,axiom,
! [A] :
( m1_subset_1(A,k16_substut1)
=> ( ~ ( v4_substut1(A)
& v5_substut1(A) )
& ~ ( v4_substut1(A)
& v6_substut1(A) )
& ~ ( v4_substut1(A)
& v7_substut1(A) )
& ~ ( v5_substut1(A)
& v6_substut1(A) )
& ~ ( v5_substut1(A)
& v7_substut1(A) )
& ~ ( v6_substut1(A)
& v7_substut1(A) )
& ~ ( v2_substut1(A)
& v4_substut1(A) )
& ~ ( v2_substut1(A)
& v5_substut1(A) )
& ~ ( v2_substut1(A)
& v6_substut1(A) )
& ~ ( v2_substut1(A)
& v7_substut1(A) ) ) ) ).
fof(d35_substut1,axiom,
! [A] :
( m1_subset_1(A,k16_substut1)
=> k32_substut1(A) = A ) ).
fof(d36_substut1,axiom,
! [A] :
( m1_subset_1(A,k2_zfmisc_1(k8_qc_lang1,k1_substut1))
=> ( ( r2_hidden(k20_qc_lang1(k33_substut1(A)),k2_relat_1(k7_substut1(k20_qc_lang1(k33_substut1(A)),k33_substut1(A),k34_substut1(A))))
=> k35_substut1(A) = k2_qc_lang3(k13_substut1(k7_substut1(k20_qc_lang1(k33_substut1(A)),k33_substut1(A),k34_substut1(A)),k21_qc_lang1(k33_substut1(A)))) )
& ( ~ r2_hidden(k20_qc_lang1(k33_substut1(A)),k2_relat_1(k7_substut1(k20_qc_lang1(k33_substut1(A)),k33_substut1(A),k34_substut1(A))))
=> k35_substut1(A) = k20_qc_lang1(k33_substut1(A)) ) ) ) ).
fof(d37_substut1,axiom,
! [A] :
( m1_subset_1(A,k16_substut1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> k36_substut1(A,B) = k14_qc_lang1(k35_substut1(k32_substut1(A)),B) ) ) ).
fof(d38_substut1,axiom,
! [A] :
( m1_subset_1(A,k16_substut1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> ( B = k37_substut1(A)
<=> ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,k16_substut1,k8_qc_lang1)
& m2_relset_1(C,k16_substut1,k8_qc_lang1)
& B = k8_funct_2(k16_substut1,k8_qc_lang1,C,A)
& ! [D] :
( m1_subset_1(D,k16_substut1)
=> ( ( v2_substut1(D)
=> k8_funct_2(k16_substut1,k8_qc_lang1,C,D) = k9_cqc_lang )
& ( v4_substut1(D)
=> k8_funct_2(k16_substut1,k8_qc_lang1,C,D) = k9_qc_lang1(k15_qc_lang1(k18_substut1(D)),k3_substut1(k26_substut1(D),k19_substut1(D))) )
& ( v5_substut1(D)
=> k8_funct_2(k16_substut1,k8_qc_lang1,C,D) = k12_qc_lang1(k8_funct_2(k16_substut1,k8_qc_lang1,C,k27_substut1(D))) )
& ( v6_substut1(D)
=> k8_funct_2(k16_substut1,k8_qc_lang1,C,D) = k13_qc_lang1(k8_funct_2(k16_substut1,k8_qc_lang1,C,k28_substut1(D)),k8_funct_2(k16_substut1,k8_qc_lang1,C,k29_substut1(D))) )
& ( v7_substut1(D)
=> k8_funct_2(k16_substut1,k8_qc_lang1,C,D) = k36_substut1(D,k8_funct_2(k16_substut1,k8_qc_lang1,C,k31_substut1(D))) ) ) ) ) ) ) ) ).
fof(t28_substut1,axiom,
! [A] :
( m1_subset_1(A,k16_substut1)
=> ( v5_substut1(A)
=> k37_substut1(A) = k12_qc_lang1(k37_substut1(k27_substut1(A))) ) ) ).
fof(t29_substut1,axiom,
! [A] :
( m1_subset_1(A,k16_substut1)
=> k37_substut1(k20_substut1(A)) = k12_qc_lang1(k37_substut1(A)) ) ).
fof(t30_substut1,axiom,
! [A] :
( m1_subset_1(A,k16_substut1)
=> ( v6_substut1(A)
=> k37_substut1(A) = k13_qc_lang1(k37_substut1(k28_substut1(A)),k37_substut1(k29_substut1(A))) ) ) ).
fof(t31_substut1,axiom,
! [A] :
( m1_subset_1(A,k16_substut1)
=> ! [B] :
( m1_subset_1(B,k16_substut1)
=> ( k19_substut1(A) = k19_substut1(B)
=> k37_substut1(k21_substut1(A,B)) = k13_qc_lang1(k37_substut1(A),k37_substut1(B)) ) ) ) ).
fof(t32_substut1,axiom,
! [A] :
( m1_subset_1(A,k16_substut1)
=> ( v7_substut1(A)
=> k37_substut1(A) = k36_substut1(A,k37_substut1(k31_substut1(A))) ) ) ).
fof(t33_substut1,axiom,
! [A] :
( m1_subset_1(A,k16_substut1)
=> ( v2_substut1(A)
=> m2_subset_1(k37_substut1(A),k8_qc_lang1,k7_cqc_lang) ) ) ).
fof(t34_substut1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ( ( v1_cqc_lang(B,A)
& m1_qc_lang1(B,A) )
<=> ( m2_finseq_1(B,k2_qc_lang1)
& k3_finseq_1(B) = A ) ) ) ) ).
fof(t35_substut1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k5_qc_lang1,k7_qc_lang1(A))
=> ! [C] :
( ( v1_cqc_lang(C,A)
& m1_qc_lang1(C,A) )
=> ! [D] :
( m1_subset_1(D,k1_substut1)
=> m2_subset_1(k37_substut1(k17_substut1(B,C,D)),k8_qc_lang1,k7_cqc_lang) ) ) ) ) ).
fof(t36_substut1,axiom,
! [A] :
( m1_subset_1(A,k16_substut1)
=> ( m2_subset_1(k37_substut1(A),k8_qc_lang1,k7_cqc_lang)
=> m2_subset_1(k37_substut1(k20_substut1(A)),k8_qc_lang1,k7_cqc_lang) ) ) ).
fof(t37_substut1,axiom,
! [A] :
( m1_subset_1(A,k16_substut1)
=> ! [B] :
( m1_subset_1(B,k16_substut1)
=> ( ( k19_substut1(A) = k19_substut1(B)
& m2_subset_1(k37_substut1(A),k8_qc_lang1,k7_cqc_lang)
& m2_subset_1(k37_substut1(B),k8_qc_lang1,k7_cqc_lang) )
=> m2_subset_1(k37_substut1(k21_substut1(A,B)),k8_qc_lang1,k7_cqc_lang) ) ) ) ).
fof(t38_substut1,axiom,
! [A] :
( m2_subset_1(A,k1_qc_lang1,k2_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k16_substut1)
=> ! [C] :
( m1_substut1(C,k25_substut1(B,A))
=> ( ( m2_subset_1(k37_substut1(B),k8_qc_lang1,k7_cqc_lang)
& v3_substut1(k25_substut1(B,A)) )
=> m2_subset_1(k37_substut1(k24_substut1(k25_substut1(B,A),C)),k8_qc_lang1,k7_cqc_lang) ) ) ) ) ).
fof(s1_substut1,axiom,
( ( ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k5_qc_lang1,k7_qc_lang1(A))
=> ! [C] :
( m1_qc_lang1(C,A)
=> ! [D] :
( m1_subset_1(D,k1_substut1)
=> p1_s1_substut1(k17_substut1(B,C,D)) ) ) ) )
& ! [A] :
( m1_subset_1(A,k16_substut1)
=> ( v2_substut1(A)
=> p1_s1_substut1(A) ) )
& ! [A] :
( m1_subset_1(A,k16_substut1)
=> ( p1_s1_substut1(A)
=> p1_s1_substut1(k20_substut1(A)) ) )
& ! [A] :
( m1_subset_1(A,k16_substut1)
=> ! [B] :
( m1_subset_1(B,k16_substut1)
=> ( ( k19_substut1(A) = k19_substut1(B)
& p1_s1_substut1(A)
& p1_s1_substut1(B) )
=> p1_s1_substut1(k21_substut1(A,B)) ) ) )
& ! [A] :
( m2_subset_1(A,k1_qc_lang1,k2_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k16_substut1)
=> ! [C] :
( m1_substut1(C,k25_substut1(B,A))
=> ( ( v3_substut1(k25_substut1(B,A))
& p1_s1_substut1(B) )
=> p1_s1_substut1(k24_substut1(k25_substut1(B,A),C)) ) ) ) ) )
=> ! [A] :
( m1_subset_1(A,k16_substut1)
=> p1_s1_substut1(A) ) ) ).
fof(s2_substut1,axiom,
( ! [A] :
( m1_subset_1(A,k16_substut1)
=> ( ( v4_substut1(A)
=> p1_s2_substut1(A) )
& ( v2_substut1(A)
=> p1_s2_substut1(A) )
& ( ( v5_substut1(A)
& p1_s2_substut1(k27_substut1(A)) )
=> p1_s2_substut1(A) )
& ( ( v6_substut1(A)
& p1_s2_substut1(k28_substut1(A))
& p1_s2_substut1(k29_substut1(A)) )
=> p1_s2_substut1(A) )
& ( ( v7_substut1(A)
& p1_s2_substut1(k31_substut1(A)) )
=> p1_s2_substut1(A) ) ) )
=> ! [A] :
( m1_subset_1(A,k16_substut1)
=> p1_s2_substut1(A) ) ) ).
fof(s3_substut1,axiom,
? [A] :
( v1_funct_1(A)
& v1_funct_2(A,k16_substut1,f1_s3_substut1)
& m2_relset_1(A,k16_substut1,f1_s3_substut1)
& ! [B] :
( m1_subset_1(B,k16_substut1)
=> ! [C] :
( m1_subset_1(C,f1_s3_substut1)
=> ! [D] :
( m1_subset_1(D,f1_s3_substut1)
=> ( ( v2_substut1(B)
=> k8_funct_2(k16_substut1,f1_s3_substut1,A,B) = f2_s3_substut1 )
& ( v4_substut1(B)
=> k8_funct_2(k16_substut1,f1_s3_substut1,A,B) = f3_s3_substut1(B) )
& ( ( v5_substut1(B)
& C = k8_funct_2(k16_substut1,f1_s3_substut1,A,k27_substut1(B)) )
=> k8_funct_2(k16_substut1,f1_s3_substut1,A,B) = f4_s3_substut1(C) )
& ( ( v6_substut1(B)
& C = k8_funct_2(k16_substut1,f1_s3_substut1,A,k28_substut1(B))
& D = k8_funct_2(k16_substut1,f1_s3_substut1,A,k29_substut1(B)) )
=> k8_funct_2(k16_substut1,f1_s3_substut1,A,B) = f5_s3_substut1(C,D) )
& ( ( v7_substut1(B)
& C = k8_funct_2(k16_substut1,f1_s3_substut1,A,k31_substut1(B)) )
=> k8_funct_2(k16_substut1,f1_s3_substut1,A,B) = f6_s3_substut1(B,C) ) ) ) ) ) ) ).
fof(s4_substut1,axiom,
( ( ! [A] :
( m1_subset_1(A,k16_substut1)
=> ! [B] :
( m1_subset_1(B,f1_s4_substut1)
=> ! [C] :
( m1_subset_1(C,f1_s4_substut1)
=> ( ( v2_substut1(A)
=> k8_funct_2(k16_substut1,f1_s4_substut1,f2_s4_substut1,A) = f4_s4_substut1 )
& ( v4_substut1(A)
=> k8_funct_2(k16_substut1,f1_s4_substut1,f2_s4_substut1,A) = f5_s4_substut1(A) )
& ( ( v5_substut1(A)
& B = k8_funct_2(k16_substut1,f1_s4_substut1,f2_s4_substut1,k27_substut1(A)) )
=> k8_funct_2(k16_substut1,f1_s4_substut1,f2_s4_substut1,A) = f6_s4_substut1(B) )
& ( ( v6_substut1(A)
& B = k8_funct_2(k16_substut1,f1_s4_substut1,f2_s4_substut1,k28_substut1(A))
& C = k8_funct_2(k16_substut1,f1_s4_substut1,f2_s4_substut1,k29_substut1(A)) )
=> k8_funct_2(k16_substut1,f1_s4_substut1,f2_s4_substut1,A) = f7_s4_substut1(B,C) )
& ( ( v7_substut1(A)
& B = k8_funct_2(k16_substut1,f1_s4_substut1,f2_s4_substut1,k31_substut1(A)) )
=> k8_funct_2(k16_substut1,f1_s4_substut1,f2_s4_substut1,A) = f8_s4_substut1(A,B) ) ) ) ) )
& ! [A] :
( m1_subset_1(A,k16_substut1)
=> ! [B] :
( m1_subset_1(B,f1_s4_substut1)
=> ! [C] :
( m1_subset_1(C,f1_s4_substut1)
=> ( ( v2_substut1(A)
=> k8_funct_2(k16_substut1,f1_s4_substut1,f3_s4_substut1,A) = f4_s4_substut1 )
& ( v4_substut1(A)
=> k8_funct_2(k16_substut1,f1_s4_substut1,f3_s4_substut1,A) = f5_s4_substut1(A) )
& ( ( v5_substut1(A)
& B = k8_funct_2(k16_substut1,f1_s4_substut1,f3_s4_substut1,k27_substut1(A)) )
=> k8_funct_2(k16_substut1,f1_s4_substut1,f3_s4_substut1,A) = f6_s4_substut1(B) )
& ( ( v6_substut1(A)
& B = k8_funct_2(k16_substut1,f1_s4_substut1,f3_s4_substut1,k28_substut1(A))
& C = k8_funct_2(k16_substut1,f1_s4_substut1,f3_s4_substut1,k29_substut1(A)) )
=> k8_funct_2(k16_substut1,f1_s4_substut1,f3_s4_substut1,A) = f7_s4_substut1(B,C) )
& ( ( v7_substut1(A)
& B = k8_funct_2(k16_substut1,f1_s4_substut1,f3_s4_substut1,k31_substut1(A)) )
=> k8_funct_2(k16_substut1,f1_s4_substut1,f3_s4_substut1,A) = f8_s4_substut1(A,B) ) ) ) ) ) )
=> f2_s4_substut1 = f3_s4_substut1 ) ).
fof(s5_substut1,axiom,
( ! [A] :
( m2_subset_1(A,k16_substut1,k38_substut1)
=> ! [B] :
( m2_subset_1(B,k16_substut1,k38_substut1)
=> ! [C] :
( m2_subset_1(C,k1_qc_lang1,k2_qc_lang1)
=> ! [D] :
( m1_substut1(D,k25_substut1(A,C))
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ! [F] :
( ( v1_cqc_lang(F,E)
& m1_qc_lang1(F,E) )
=> ! [G] :
( m2_subset_1(G,k5_qc_lang1,k7_qc_lang1(E))
=> ! [H] :
( m1_subset_1(H,k1_substut1)
=> ( p1_s5_substut1(k17_substut1(G,F,H))
& ( v2_substut1(A)
=> p1_s5_substut1(A) )
& ( p1_s5_substut1(A)
=> p1_s5_substut1(k20_substut1(A)) )
& ( ( k19_substut1(A) = k19_substut1(B)
& p1_s5_substut1(A)
& p1_s5_substut1(B) )
=> p1_s5_substut1(k21_substut1(A,B)) )
& ( ( v3_substut1(k25_substut1(A,C))
& p1_s5_substut1(A) )
=> p1_s5_substut1(k24_substut1(k25_substut1(A,C),D)) ) ) ) ) ) ) ) ) ) )
=> ! [A] :
( m2_subset_1(A,k16_substut1,k38_substut1)
=> p1_s5_substut1(A) ) ) ).
fof(dt_m1_substut1,axiom,
! [A] :
( m1_subset_1(A,k2_zfmisc_1(k16_substut1,k2_qc_lang1))
=> ! [B] :
( m1_substut1(B,A)
=> m1_subset_1(B,k1_substut1) ) ) ).
fof(existence_m1_substut1,axiom,
! [A] :
( m1_subset_1(A,k2_zfmisc_1(k16_substut1,k2_qc_lang1))
=> ? [B] : m1_substut1(B,A) ) ).
fof(dt_k1_substut1,axiom,
$true ).
fof(dt_k2_substut1,axiom,
! [A] :
( m1_subset_1(A,k1_substut1)
=> ( v1_funct_1(k2_substut1(A))
& m2_relset_1(k2_substut1(A),k2_qc_lang1,k2_qc_lang1) ) ) ).
fof(dt_k3_substut1,axiom,
! [A,B] :
( ( m1_finseq_1(A,k1_qc_lang1)
& m1_subset_1(B,k1_substut1) )
=> m2_finseq_1(k3_substut1(A,B),k1_qc_lang1) ) ).
fof(dt_k4_substut1,axiom,
! [A] :
( m1_finseq_1(A,k2_qc_lang1)
=> m2_finseq_1(k4_substut1(A),k1_qc_lang1) ) ).
fof(dt_k5_substut1,axiom,
! [A,B] :
( ( m1_finseq_1(A,k2_qc_lang1)
& m1_subset_1(B,k1_substut1) )
=> m2_finseq_1(k5_substut1(A,B),k2_qc_lang1) ) ).
fof(dt_k6_substut1,axiom,
! [A,B] :
( m1_subset_1(A,k1_substut1)
=> m1_subset_1(k6_substut1(A,B),k1_substut1) ) ).
fof(redefinition_k6_substut1,axiom,
! [A,B] :
( m1_subset_1(A,k1_substut1)
=> k6_substut1(A,B) = k7_relat_1(A,B) ) ).
fof(dt_k7_substut1,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k2_qc_lang1)
& m1_subset_1(B,k8_qc_lang1)
& m1_subset_1(C,k1_substut1) )
=> ( v1_finset_1(k7_substut1(A,B,C))
& m1_subset_1(k7_substut1(A,B,C),k1_substut1) ) ) ).
fof(dt_k8_substut1,axiom,
! [A] :
( m1_finseq_1(A,k1_qc_lang1)
=> m1_subset_1(k8_substut1(A),k1_zfmisc_1(k2_qc_lang1)) ) ).
fof(dt_k9_substut1,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> m1_subset_1(k9_substut1(A),k1_zfmisc_1(k2_qc_lang1)) ) ).
fof(dt_k10_substut1,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ( v1_finset_1(k10_substut1(A))
& m1_subset_1(k10_substut1(A),k1_zfmisc_1(k5_numbers)) ) ) ).
fof(dt_k11_substut1,axiom,
! [A] :
( ( v1_finset_1(A)
& m1_subset_1(A,k1_substut1) )
=> ( v1_finset_1(k11_substut1(A))
& m1_subset_1(k11_substut1(A),k1_zfmisc_1(k5_numbers)) ) ) ).
fof(dt_k12_substut1,axiom,
! [A,B] :
( ( m1_subset_1(A,k8_qc_lang1)
& v1_finset_1(B)
& m1_subset_1(B,k1_substut1) )
=> ( ~ v1_xboole_0(k12_substut1(A,B))
& m1_subset_1(k12_substut1(A,B),k1_zfmisc_1(k5_numbers)) ) ) ).
fof(dt_k13_substut1,axiom,
! [A,B] :
( ( v1_finset_1(A)
& m1_subset_1(A,k1_substut1)
& m1_subset_1(B,k8_qc_lang1) )
=> m2_subset_1(k13_substut1(A,B),k1_numbers,k5_numbers) ) ).
fof(dt_k14_substut1,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k2_qc_lang1)
& m1_subset_1(B,k8_qc_lang1)
& v1_finset_1(C)
& m1_subset_1(C,k1_substut1) )
=> m1_subset_1(k14_substut1(A,B,C),k1_substut1) ) ).
fof(dt_k15_substut1,axiom,
( v1_relat_1(k15_substut1)
& v1_funct_1(k15_substut1) ) ).
fof(dt_k16_substut1,axiom,
~ v1_xboole_0(k16_substut1) ).
fof(dt_k17_substut1,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_qc_lang1)
& m1_finseq_1(B,k1_qc_lang1)
& m1_subset_1(C,k1_substut1) )
=> m1_subset_1(k17_substut1(A,B,C),k16_substut1) ) ).
fof(dt_k18_substut1,axiom,
! [A] :
( m1_subset_1(A,k16_substut1)
=> m1_subset_1(k18_substut1(A),k8_qc_lang1) ) ).
fof(redefinition_k18_substut1,axiom,
! [A] :
( m1_subset_1(A,k16_substut1)
=> k18_substut1(A) = k1_mcart_1(A) ) ).
fof(dt_k19_substut1,axiom,
! [A] :
( m1_subset_1(A,k16_substut1)
=> m1_subset_1(k19_substut1(A),k1_substut1) ) ).
fof(redefinition_k19_substut1,axiom,
! [A] :
( m1_subset_1(A,k16_substut1)
=> k19_substut1(A) = k2_mcart_1(A) ) ).
fof(dt_k20_substut1,axiom,
! [A] :
( m1_subset_1(A,k16_substut1)
=> m1_subset_1(k20_substut1(A),k16_substut1) ) ).
fof(dt_k21_substut1,axiom,
! [A,B] :
( ( m1_subset_1(A,k16_substut1)
& m1_subset_1(B,k16_substut1) )
=> m1_subset_1(k21_substut1(A,B),k16_substut1) ) ).
fof(dt_k22_substut1,axiom,
! [A] :
( m1_subset_1(A,k2_zfmisc_1(k16_substut1,k2_qc_lang1))
=> m1_subset_1(k22_substut1(A),k16_substut1) ) ).
fof(redefinition_k22_substut1,axiom,
! [A] :
( m1_subset_1(A,k2_zfmisc_1(k16_substut1,k2_qc_lang1))
=> k22_substut1(A) = k1_mcart_1(A) ) ).
fof(dt_k23_substut1,axiom,
! [A] :
( m1_subset_1(A,k2_zfmisc_1(k16_substut1,k2_qc_lang1))
=> m2_subset_1(k23_substut1(A),k1_qc_lang1,k2_qc_lang1) ) ).
fof(redefinition_k23_substut1,axiom,
! [A] :
( m1_subset_1(A,k2_zfmisc_1(k16_substut1,k2_qc_lang1))
=> k23_substut1(A) = k2_mcart_1(A) ) ).
fof(dt_k24_substut1,axiom,
! [A,B] :
( ( m1_subset_1(A,k2_zfmisc_1(k16_substut1,k2_qc_lang1))
& m1_substut1(B,A) )
=> m1_subset_1(k24_substut1(A,B),k16_substut1) ) ).
fof(dt_k25_substut1,axiom,
! [A,B] :
( ( m1_subset_1(A,k16_substut1)
& m1_subset_1(B,k2_qc_lang1) )
=> m1_subset_1(k25_substut1(A,B),k2_zfmisc_1(k16_substut1,k2_qc_lang1)) ) ).
fof(redefinition_k25_substut1,axiom,
! [A,B] :
( ( m1_subset_1(A,k16_substut1)
& m1_subset_1(B,k2_qc_lang1) )
=> k25_substut1(A,B) = k4_tarski(A,B) ) ).
fof(dt_k26_substut1,axiom,
! [A] :
( m1_subset_1(A,k16_substut1)
=> m2_finseq_1(k26_substut1(A),k1_qc_lang1) ) ).
fof(dt_k27_substut1,axiom,
! [A] :
( m1_subset_1(A,k16_substut1)
=> m1_subset_1(k27_substut1(A),k16_substut1) ) ).
fof(dt_k28_substut1,axiom,
! [A] :
( m1_subset_1(A,k16_substut1)
=> m1_subset_1(k28_substut1(A),k16_substut1) ) ).
fof(dt_k29_substut1,axiom,
! [A] :
( m1_subset_1(A,k16_substut1)
=> m1_subset_1(k29_substut1(A),k16_substut1) ) ).
fof(dt_k30_substut1,axiom,
! [A] : m2_subset_1(k30_substut1(A),k1_qc_lang1,k2_qc_lang1) ).
fof(dt_k31_substut1,axiom,
! [A] : m1_subset_1(k31_substut1(A),k16_substut1) ).
fof(dt_k32_substut1,axiom,
! [A] :
( m1_subset_1(A,k16_substut1)
=> m1_subset_1(k32_substut1(A),k2_zfmisc_1(k8_qc_lang1,k1_substut1)) ) ).
fof(dt_k33_substut1,axiom,
! [A] :
( m1_subset_1(A,k2_zfmisc_1(k8_qc_lang1,k1_substut1))
=> m1_subset_1(k33_substut1(A),k8_qc_lang1) ) ).
fof(redefinition_k33_substut1,axiom,
! [A] :
( m1_subset_1(A,k2_zfmisc_1(k8_qc_lang1,k1_substut1))
=> k33_substut1(A) = k1_mcart_1(A) ) ).
fof(dt_k34_substut1,axiom,
! [A] :
( m1_subset_1(A,k2_zfmisc_1(k8_qc_lang1,k1_substut1))
=> m1_subset_1(k34_substut1(A),k1_substut1) ) ).
fof(redefinition_k34_substut1,axiom,
! [A] :
( m1_subset_1(A,k2_zfmisc_1(k8_qc_lang1,k1_substut1))
=> k34_substut1(A) = k2_mcart_1(A) ) ).
fof(dt_k35_substut1,axiom,
! [A] :
( m1_subset_1(A,k2_zfmisc_1(k8_qc_lang1,k1_substut1))
=> m2_subset_1(k35_substut1(A),k1_qc_lang1,k2_qc_lang1) ) ).
fof(dt_k36_substut1,axiom,
! [A,B] :
( ( m1_subset_1(A,k16_substut1)
& m1_subset_1(B,k8_qc_lang1) )
=> m1_subset_1(k36_substut1(A,B),k8_qc_lang1) ) ).
fof(dt_k37_substut1,axiom,
! [A] :
( m1_subset_1(A,k16_substut1)
=> m1_subset_1(k37_substut1(A),k8_qc_lang1) ) ).
fof(dt_k38_substut1,axiom,
m1_subset_1(k38_substut1,k1_zfmisc_1(k16_substut1)) ).
fof(dt_k39_substut1,axiom,
! [A] :
( m1_subset_1(A,k38_substut1)
=> m2_subset_1(k39_substut1(A),k8_qc_lang1,k7_cqc_lang) ) ).
fof(redefinition_k39_substut1,axiom,
! [A] :
( m1_subset_1(A,k38_substut1)
=> k39_substut1(A) = k37_substut1(A) ) ).
fof(d6_substut1,axiom,
! [A] :
( m2_subset_1(A,k1_qc_lang1,k2_qc_lang1)
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> ! [C] :
( m1_subset_1(C,k1_substut1)
=> k7_substut1(A,B,C) = k6_substut1(C,a_3_0_substut1(A,B,C)) ) ) ) ).
fof(d7_substut1,axiom,
! [A] :
( m2_finseq_1(A,k1_qc_lang1)
=> k8_substut1(A) = a_1_0_substut1(A) ) ).
fof(d9_substut1,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> k10_substut1(A) = a_1_1_substut1(A) ) ).
fof(d10_substut1,axiom,
! [A] :
( ( v1_finset_1(A)
& m1_subset_1(A,k1_substut1) )
=> k11_substut1(A) = a_1_2_substut1(A) ) ).
fof(d39_substut1,axiom,
k38_substut1 = a_0_0_substut1 ).
fof(fraenkel_a_3_0_substut1,axiom,
! [A,B,C,D] :
( ( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1)
& m1_subset_1(C,k8_qc_lang1)
& m1_subset_1(D,k1_substut1) )
=> ( r2_hidden(A,a_3_0_substut1(B,C,D))
<=> ? [E] :
( m2_subset_1(E,k1_qc_lang1,k2_qc_lang1)
& A = E
& r2_hidden(E,k24_qc_lang1(C))
& m1_subset_1(E,k1_relat_1(D))
& E != B
& E != k1_funct_1(D,E) ) ) ) ).
fof(fraenkel_a_1_0_substut1,axiom,
! [A,B] :
( m2_finseq_1(B,k1_qc_lang1)
=> ( r2_hidden(A,a_1_0_substut1(B))
<=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& A = k1_funct_1(B,C)
& r1_xreal_0(np__1,C)
& r1_xreal_0(C,k3_finseq_1(B))
& r2_hidden(k1_funct_1(B,C),k2_qc_lang1) ) ) ) ).
fof(fraenkel_a_1_1_substut1,axiom,
! [A,B] :
( m1_subset_1(B,k8_qc_lang1)
=> ( r2_hidden(A,a_1_1_substut1(B))
<=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& A = C
& r2_hidden(k2_qc_lang3(C),k9_substut1(B)) ) ) ) ).
fof(fraenkel_a_1_2_substut1,axiom,
! [A,B] :
( ( v1_finset_1(B)
& m1_subset_1(B,k1_substut1) )
=> ( r2_hidden(A,a_1_2_substut1(B))
<=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& A = C
& r2_hidden(k2_qc_lang3(C),k2_relat_1(B)) ) ) ) ).
fof(fraenkel_a_0_0_substut1,axiom,
! [A] :
( r2_hidden(A,a_0_0_substut1)
<=> ? [B] :
( m1_subset_1(B,k16_substut1)
& A = B
& m2_subset_1(k18_substut1(B),k8_qc_lang1,k7_cqc_lang) ) ) ).
%------------------------------------------------------------------------------