SET007 Axioms: SET007+101.ax
%------------------------------------------------------------------------------
% File : SET007+101 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : A First-Order Predicate Calculus
% Version : [Urb08] axioms.
% English : Axiomatics, the Consequence Operation and a Concept of Proof
% 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 : cqc_the1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 127 ( 40 unt; 0 def)
% Number of atoms : 570 ( 112 equ)
% Maximal formula atoms : 71 ( 4 avg)
% Number of connectives : 541 ( 98 ~; 10 |; 190 &)
% ( 24 <=>; 219 =>; 0 <=; 0 <~>)
% Maximal formula depth : 32 ( 6 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 17 ( 15 usr; 1 prp; 0-3 aty)
% Number of functors : 45 ( 45 usr; 21 con; 0-3 aty)
% Number of variables : 219 ( 191 !; 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(fc1_cqc_the1,axiom,
~ v1_xboole_0(k2_cqc_the1) ).
fof(t1_cqc_the1,axiom,
$true ).
fof(t2_cqc_the1,axiom,
! [A] :
( v4_ordinal2(A)
=> ~ ( r1_xreal_0(A,np__1)
& A != np__0
& A != np__1 ) ) ).
fof(t3_cqc_the1,axiom,
! [A] :
( v4_ordinal2(A)
=> ~ ( r1_xreal_0(A,np__2)
& A != np__0
& A != np__1
& A != np__2 ) ) ).
fof(t4_cqc_the1,axiom,
! [A] :
( v4_ordinal2(A)
=> ~ ( r1_xreal_0(A,np__3)
& A != np__0
& A != np__1
& A != np__2
& A != np__3 ) ) ).
fof(t5_cqc_the1,axiom,
! [A] :
( v4_ordinal2(A)
=> ~ ( r1_xreal_0(A,np__4)
& A != np__0
& A != np__1
& A != np__2
& A != np__3
& A != np__4 ) ) ).
fof(t6_cqc_the1,axiom,
! [A] :
( v4_ordinal2(A)
=> ~ ( r1_xreal_0(A,np__5)
& A != np__0
& A != np__1
& A != np__2
& A != np__3
& A != np__4
& A != np__5 ) ) ).
fof(t7_cqc_the1,axiom,
! [A] :
( v4_ordinal2(A)
=> ~ ( r1_xreal_0(A,np__6)
& A != np__0
& A != np__1
& A != np__2
& A != np__3
& A != np__4
& A != np__5
& A != np__6 ) ) ).
fof(t8_cqc_the1,axiom,
! [A] :
( v4_ordinal2(A)
=> ~ ( r1_xreal_0(A,np__7)
& A != np__0
& A != np__1
& A != np__2
& A != np__3
& A != np__4
& A != np__5
& A != np__6
& A != np__7 ) ) ).
fof(t9_cqc_the1,axiom,
! [A] :
( v4_ordinal2(A)
=> ~ ( r1_xreal_0(A,np__8)
& A != np__0
& A != np__1
& A != np__2
& A != np__3
& A != np__4
& A != np__5
& A != np__6
& A != np__7
& A != np__8 ) ) ).
fof(t10_cqc_the1,axiom,
! [A] :
( v4_ordinal2(A)
=> ~ ( r1_xreal_0(A,np__9)
& A != np__0
& A != np__1
& A != np__2
& A != np__3
& A != np__4
& A != np__5
& A != np__6
& A != np__7
& A != np__8
& A != np__9 ) ) ).
fof(t13_cqc_the1,axiom,
! [A,B,C] :
~ ( v1_finset_1(A)
& r1_tarski(A,k2_zfmisc_1(B,C))
& ! [D,E] :
~ ( v1_finset_1(D)
& r1_tarski(D,B)
& v1_finset_1(E)
& r1_tarski(E,C)
& r1_tarski(A,k2_zfmisc_1(D,E)) ) ) ).
fof(t14_cqc_the1,axiom,
! [A,B,C] :
~ ( v1_finset_1(A)
& v1_finset_1(B)
& r1_tarski(A,k2_zfmisc_1(C,B))
& ! [D] :
~ ( v1_finset_1(D)
& r1_tarski(D,C)
& r1_tarski(A,k2_zfmisc_1(D,B)) ) ) ).
fof(d1_cqc_the1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> ( v1_cqc_the1(A)
<=> ( r2_hidden(k9_cqc_lang,A)
& ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ! [C] :
( m2_subset_1(C,k8_qc_lang1,k7_cqc_lang)
=> ! [D] :
( m2_subset_1(D,k8_qc_lang1,k7_cqc_lang)
=> ! [E] :
( m1_subset_1(E,k8_qc_lang1)
=> ! [F] :
( m2_subset_1(F,k1_qc_lang1,k2_qc_lang1)
=> ! [G] :
( m2_subset_1(G,k1_qc_lang1,k2_qc_lang1)
=> ( r2_hidden(k12_cqc_lang(k12_cqc_lang(k10_cqc_lang(B),B),B),A)
& r2_hidden(k12_cqc_lang(B,k12_cqc_lang(k10_cqc_lang(B),C)),A)
& r2_hidden(k12_cqc_lang(k12_cqc_lang(B,C),k12_cqc_lang(k10_cqc_lang(k11_cqc_lang(C,D)),k10_cqc_lang(k11_cqc_lang(B,D)))),A)
& r2_hidden(k12_cqc_lang(k11_cqc_lang(B,C),k11_cqc_lang(C,B)),A)
& ( ( r2_hidden(B,A)
& r2_hidden(k12_cqc_lang(B,C),A) )
=> r2_hidden(C,A) )
& r2_hidden(k12_cqc_lang(k15_cqc_lang(F,B),B),A)
& ( r2_hidden(k12_cqc_lang(B,C),A)
=> ( r2_hidden(F,k24_qc_lang1(B))
| r2_hidden(k12_cqc_lang(B,k15_cqc_lang(F,C)),A) ) )
& ( ( r2_hidden(k17_cqc_lang(E,F),k7_cqc_lang)
& r2_hidden(k17_cqc_lang(E,G),k7_cqc_lang)
& r2_hidden(k17_cqc_lang(E,F),A) )
=> ( r2_hidden(F,k24_qc_lang1(E))
| r2_hidden(k17_cqc_lang(E,G),A) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t15_cqc_the1,axiom,
$true ).
fof(t16_cqc_the1,axiom,
$true ).
fof(t17_cqc_the1,axiom,
$true ).
fof(t18_cqc_the1,axiom,
$true ).
fof(t19_cqc_the1,axiom,
$true ).
fof(t20_cqc_the1,axiom,
$true ).
fof(t21_cqc_the1,axiom,
$true ).
fof(t22_cqc_the1,axiom,
$true ).
fof(t23_cqc_the1,axiom,
$true ).
fof(t24_cqc_the1,axiom,
$true ).
fof(t25_cqc_the1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k7_cqc_lang))
=> ( ( v1_cqc_the1(A)
& v1_cqc_the1(B) )
=> v1_cqc_the1(k5_subset_1(k7_cqc_lang,A,B)) ) ) ) ).
fof(d2_cqc_the1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k7_cqc_lang))
=> ( B = k1_cqc_the1(A)
<=> ! [C] :
( m2_subset_1(C,k8_qc_lang1,k7_cqc_lang)
=> ( r2_hidden(C,B)
<=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k7_cqc_lang))
=> ( ( v1_cqc_the1(D)
& r1_tarski(A,D) )
=> r2_hidden(C,D) ) ) ) ) ) ) ) ).
fof(t26_cqc_the1,axiom,
$true ).
fof(t27_cqc_the1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> r2_hidden(k9_cqc_lang,k1_cqc_the1(A)) ) ).
fof(t28_cqc_the1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> r2_hidden(k12_cqc_lang(k12_cqc_lang(k10_cqc_lang(B),B),B),k1_cqc_the1(A)) ) ) ).
fof(t29_cqc_the1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ! [C] :
( m2_subset_1(C,k8_qc_lang1,k7_cqc_lang)
=> r2_hidden(k12_cqc_lang(B,k12_cqc_lang(k10_cqc_lang(B),C)),k1_cqc_the1(A)) ) ) ) ).
fof(t30_cqc_the1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ! [C] :
( m2_subset_1(C,k8_qc_lang1,k7_cqc_lang)
=> ! [D] :
( m2_subset_1(D,k8_qc_lang1,k7_cqc_lang)
=> r2_hidden(k12_cqc_lang(k12_cqc_lang(B,C),k12_cqc_lang(k10_cqc_lang(k11_cqc_lang(C,D)),k10_cqc_lang(k11_cqc_lang(B,D)))),k1_cqc_the1(A)) ) ) ) ) ).
fof(t31_cqc_the1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ! [C] :
( m2_subset_1(C,k8_qc_lang1,k7_cqc_lang)
=> r2_hidden(k12_cqc_lang(k11_cqc_lang(B,C),k11_cqc_lang(C,B)),k1_cqc_the1(A)) ) ) ) ).
fof(t32_cqc_the1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ! [C] :
( m2_subset_1(C,k8_qc_lang1,k7_cqc_lang)
=> ( ( r2_hidden(B,k1_cqc_the1(A))
& r2_hidden(k12_cqc_lang(B,C),k1_cqc_the1(A)) )
=> r2_hidden(C,k1_cqc_the1(A)) ) ) ) ) ).
fof(t33_cqc_the1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ! [C] :
( m2_subset_1(C,k1_qc_lang1,k2_qc_lang1)
=> r2_hidden(k12_cqc_lang(k15_cqc_lang(C,B),B),k1_cqc_the1(A)) ) ) ) ).
fof(t34_cqc_the1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ! [C] :
( m2_subset_1(C,k8_qc_lang1,k7_cqc_lang)
=> ! [D] :
( m2_subset_1(D,k1_qc_lang1,k2_qc_lang1)
=> ( r2_hidden(k12_cqc_lang(B,C),k1_cqc_the1(A))
=> ( r2_hidden(D,k24_qc_lang1(B))
| r2_hidden(k12_cqc_lang(B,k15_cqc_lang(D,C)),k1_cqc_the1(A)) ) ) ) ) ) ) ).
fof(t35_cqc_the1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> ! [C] :
( m2_subset_1(C,k1_qc_lang1,k2_qc_lang1)
=> ! [D] :
( m2_subset_1(D,k1_qc_lang1,k2_qc_lang1)
=> ( ( r2_hidden(k17_cqc_lang(B,C),k7_cqc_lang)
& r2_hidden(k17_cqc_lang(B,D),k7_cqc_lang)
& r2_hidden(k17_cqc_lang(B,C),k1_cqc_the1(A)) )
=> ( r2_hidden(C,k24_qc_lang1(B))
| r2_hidden(k17_cqc_lang(B,D),k1_cqc_the1(A)) ) ) ) ) ) ) ).
fof(t36_cqc_the1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> v1_cqc_the1(k1_cqc_the1(A)) ) ).
fof(t37_cqc_the1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k7_cqc_lang))
=> ( ( v1_cqc_the1(A)
& r1_tarski(B,A) )
=> r1_tarski(k1_cqc_the1(B),A) ) ) ) ).
fof(t38_cqc_the1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> r1_tarski(A,k1_cqc_the1(A)) ) ).
fof(t39_cqc_the1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k7_cqc_lang))
=> ( r1_tarski(A,B)
=> r1_tarski(k1_cqc_the1(A),k1_cqc_the1(B)) ) ) ) ).
fof(t40_cqc_the1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> k1_cqc_the1(k1_cqc_the1(A)) = k1_cqc_the1(A) ) ).
fof(t41_cqc_the1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> ( v1_cqc_the1(A)
<=> k1_cqc_the1(A) = A ) ) ).
fof(t42_cqc_the1,axiom,
$true ).
fof(t43_cqc_the1,axiom,
( r2_hidden(np__0,k2_cqc_the1)
& r2_hidden(np__1,k2_cqc_the1)
& r2_hidden(np__2,k2_cqc_the1)
& r2_hidden(np__3,k2_cqc_the1)
& r2_hidden(np__4,k2_cqc_the1)
& r2_hidden(np__5,k2_cqc_the1)
& r2_hidden(np__6,k2_cqc_the1)
& r2_hidden(np__7,k2_cqc_the1)
& r2_hidden(np__8,k2_cqc_the1)
& r2_hidden(np__9,k2_cqc_the1) ) ).
fof(t44_cqc_the1,axiom,
v1_finset_1(k2_cqc_the1) ).
fof(t45_cqc_the1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_zfmisc_1(k7_cqc_lang,k2_cqc_the1))
=> ~ ( r1_xreal_0(np__1,A)
& r1_xreal_0(A,k3_finseq_1(B))
& k2_mcart_1(k1_funct_1(B,A)) != np__0
& k2_mcart_1(k1_funct_1(B,A)) != np__1
& k2_mcart_1(k1_funct_1(B,A)) != np__2
& k2_mcart_1(k1_funct_1(B,A)) != np__3
& k2_mcart_1(k1_funct_1(B,A)) != np__4
& k2_mcart_1(k1_funct_1(B,A)) != np__5
& k2_mcart_1(k1_funct_1(B,A)) != np__6
& k2_mcart_1(k1_funct_1(B,A)) != np__7
& k2_mcart_1(k1_funct_1(B,A)) != np__8
& k2_mcart_1(k1_funct_1(B,A)) != np__9 ) ) ) ).
fof(d4_cqc_the1,axiom,
! [A] :
( m2_finseq_1(A,k2_zfmisc_1(k7_cqc_lang,k2_cqc_the1))
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k7_cqc_lang))
=> ( ( k2_mcart_1(k1_funct_1(A,B)) = np__0
=> ( r1_cqc_the1(A,B,C)
<=> r2_hidden(k1_mcart_1(k1_funct_1(A,B)),C) ) )
& ( k2_mcart_1(k1_funct_1(A,B)) = np__1
=> ( r1_cqc_the1(A,B,C)
<=> k1_mcart_1(k1_funct_1(A,B)) = k9_cqc_lang ) )
& ( k2_mcart_1(k1_funct_1(A,B)) = np__2
=> ( r1_cqc_the1(A,B,C)
<=> ? [D] :
( m2_subset_1(D,k8_qc_lang1,k7_cqc_lang)
& k1_mcart_1(k1_funct_1(A,B)) = k12_cqc_lang(k12_cqc_lang(k10_cqc_lang(D),D),D) ) ) )
& ( k2_mcart_1(k1_funct_1(A,B)) = np__3
=> ( r1_cqc_the1(A,B,C)
<=> ? [D] :
( m2_subset_1(D,k8_qc_lang1,k7_cqc_lang)
& ? [E] :
( m2_subset_1(E,k8_qc_lang1,k7_cqc_lang)
& k1_mcart_1(k1_funct_1(A,B)) = k12_cqc_lang(D,k12_cqc_lang(k10_cqc_lang(D),E)) ) ) ) )
& ( k2_mcart_1(k1_funct_1(A,B)) = np__4
=> ( r1_cqc_the1(A,B,C)
<=> ? [D] :
( m2_subset_1(D,k8_qc_lang1,k7_cqc_lang)
& ? [E] :
( m2_subset_1(E,k8_qc_lang1,k7_cqc_lang)
& ? [F] :
( m2_subset_1(F,k8_qc_lang1,k7_cqc_lang)
& k1_mcart_1(k1_funct_1(A,B)) = k12_cqc_lang(k12_cqc_lang(D,E),k12_cqc_lang(k10_cqc_lang(k11_cqc_lang(E,F)),k10_cqc_lang(k11_cqc_lang(D,F)))) ) ) ) ) )
& ( k2_mcart_1(k1_funct_1(A,B)) = np__5
=> ( r1_cqc_the1(A,B,C)
<=> ? [D] :
( m2_subset_1(D,k8_qc_lang1,k7_cqc_lang)
& ? [E] :
( m2_subset_1(E,k8_qc_lang1,k7_cqc_lang)
& k1_mcart_1(k1_funct_1(A,B)) = k12_cqc_lang(k11_cqc_lang(D,E),k11_cqc_lang(E,D)) ) ) ) )
& ( k2_mcart_1(k1_funct_1(A,B)) = np__6
=> ( r1_cqc_the1(A,B,C)
<=> ? [D] :
( m2_subset_1(D,k8_qc_lang1,k7_cqc_lang)
& ? [E] :
( m2_subset_1(E,k1_qc_lang1,k2_qc_lang1)
& k1_mcart_1(k1_funct_1(A,B)) = k12_cqc_lang(k15_cqc_lang(E,D),D) ) ) ) )
& ( k2_mcart_1(k1_funct_1(A,B)) = np__7
=> ( r1_cqc_the1(A,B,C)
<=> ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& ? [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
& ? [F] :
( m2_subset_1(F,k8_qc_lang1,k7_cqc_lang)
& ? [G] :
( m2_subset_1(G,k8_qc_lang1,k7_cqc_lang)
& r1_xreal_0(np__1,D)
& ~ r1_xreal_0(B,D)
& r1_xreal_0(np__1,E)
& ~ r1_xreal_0(D,E)
& F = k1_mcart_1(k1_funct_1(A,E))
& G = k1_mcart_1(k1_funct_1(A,B))
& k1_mcart_1(k1_funct_1(A,D)) = k12_cqc_lang(F,G) ) ) ) ) ) )
& ( k2_mcart_1(k1_funct_1(A,B)) = np__8
=> ( r1_cqc_the1(A,B,C)
<=> ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& ? [E] :
( m2_subset_1(E,k8_qc_lang1,k7_cqc_lang)
& ? [F] :
( m2_subset_1(F,k8_qc_lang1,k7_cqc_lang)
& ? [G] :
( m2_subset_1(G,k1_qc_lang1,k2_qc_lang1)
& r1_xreal_0(np__1,D)
& ~ r1_xreal_0(B,D)
& k1_mcart_1(k1_funct_1(A,D)) = k12_cqc_lang(E,F)
& ~ r2_hidden(G,k24_qc_lang1(E))
& k1_mcart_1(k1_funct_1(A,B)) = k12_cqc_lang(E,k15_cqc_lang(G,F)) ) ) ) ) ) )
& ( k2_mcart_1(k1_funct_1(A,B)) = np__9
=> ( r1_cqc_the1(A,B,C)
<=> ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& ? [E] :
( m2_subset_1(E,k1_qc_lang1,k2_qc_lang1)
& ? [F] :
( m2_subset_1(F,k1_qc_lang1,k2_qc_lang1)
& ? [G] :
( m1_subset_1(G,k8_qc_lang1)
& r1_xreal_0(np__1,D)
& ~ r1_xreal_0(B,D)
& r2_hidden(k17_cqc_lang(G,E),k7_cqc_lang)
& r2_hidden(k17_cqc_lang(G,F),k7_cqc_lang)
& ~ r2_hidden(E,k24_qc_lang1(G))
& k17_cqc_lang(G,E) = k1_mcart_1(k1_funct_1(A,D))
& k17_cqc_lang(G,F) = k1_mcart_1(k1_funct_1(A,B)) ) ) ) ) ) ) ) ) ) ) ).
fof(d5_cqc_the1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> ! [B] :
( m2_finseq_1(B,k2_zfmisc_1(k7_cqc_lang,k2_cqc_the1))
=> ( r2_cqc_the1(A,B)
<=> ( B != k1_xboole_0
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,C)
& r1_xreal_0(C,k3_finseq_1(B)) )
=> r1_cqc_the1(B,C,A) ) ) ) ) ) ) ).
fof(t46_cqc_the1,axiom,
$true ).
fof(t47_cqc_the1,axiom,
$true ).
fof(t48_cqc_the1,axiom,
$true ).
fof(t49_cqc_the1,axiom,
$true ).
fof(t50_cqc_the1,axiom,
$true ).
fof(t51_cqc_the1,axiom,
$true ).
fof(t52_cqc_the1,axiom,
$true ).
fof(t53_cqc_the1,axiom,
$true ).
fof(t54_cqc_the1,axiom,
$true ).
fof(t55_cqc_the1,axiom,
$true ).
fof(t56_cqc_the1,axiom,
$true ).
fof(t57_cqc_the1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> ! [B] :
( m2_finseq_1(B,k2_zfmisc_1(k7_cqc_lang,k2_cqc_the1))
=> ~ ( r2_cqc_the1(A,B)
& k2_relat_1(B) = k1_xboole_0 ) ) ) ).
fof(t58_cqc_the1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> ! [B] :
( m2_finseq_1(B,k2_zfmisc_1(k7_cqc_lang,k2_cqc_the1))
=> ( r2_cqc_the1(A,B)
=> r1_xreal_0(np__1,k3_finseq_1(B)) ) ) ) ).
fof(t59_cqc_the1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> ! [B] :
( m2_finseq_1(B,k2_zfmisc_1(k7_cqc_lang,k2_cqc_the1))
=> ~ ( r2_cqc_the1(A,B)
& k2_mcart_1(k1_funct_1(B,np__1)) != np__0
& k2_mcart_1(k1_funct_1(B,np__1)) != np__1
& k2_mcart_1(k1_funct_1(B,np__1)) != np__2
& k2_mcart_1(k1_funct_1(B,np__1)) != np__3
& k2_mcart_1(k1_funct_1(B,np__1)) != np__4
& k2_mcart_1(k1_funct_1(B,np__1)) != np__5
& k2_mcart_1(k1_funct_1(B,np__1)) != np__6 ) ) ) ).
fof(t60_cqc_the1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k7_cqc_lang))
=> ! [C] :
( m2_finseq_1(C,k2_zfmisc_1(k7_cqc_lang,k2_cqc_the1))
=> ! [D] :
( m2_finseq_1(D,k2_zfmisc_1(k7_cqc_lang,k2_cqc_the1))
=> ( ( r1_xreal_0(np__1,A)
& r1_xreal_0(A,k3_finseq_1(C)) )
=> ( r1_cqc_the1(C,A,B)
<=> r1_cqc_the1(k8_finseq_1(k2_zfmisc_1(k7_cqc_lang,k2_cqc_the1),C,D),A,B) ) ) ) ) ) ) ).
fof(t61_cqc_the1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k7_cqc_lang))
=> ! [C] :
( m2_finseq_1(C,k2_zfmisc_1(k7_cqc_lang,k2_cqc_the1))
=> ! [D] :
( m2_finseq_1(D,k2_zfmisc_1(k7_cqc_lang,k2_cqc_the1))
=> ( ( r1_xreal_0(np__1,A)
& r1_xreal_0(A,k3_finseq_1(C))
& r1_cqc_the1(C,A,B) )
=> r1_cqc_the1(k8_finseq_1(k2_zfmisc_1(k7_cqc_lang,k2_cqc_the1),D,C),k1_nat_1(A,k3_finseq_1(D)),B) ) ) ) ) ) ).
fof(t62_cqc_the1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> ! [B] :
( m2_finseq_1(B,k2_zfmisc_1(k7_cqc_lang,k2_cqc_the1))
=> ! [C] :
( m2_finseq_1(C,k2_zfmisc_1(k7_cqc_lang,k2_cqc_the1))
=> ( ( r2_cqc_the1(A,B)
& r2_cqc_the1(A,C) )
=> r2_cqc_the1(A,k8_finseq_1(k2_zfmisc_1(k7_cqc_lang,k2_cqc_the1),B,C)) ) ) ) ) ).
fof(t63_cqc_the1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k7_cqc_lang))
=> ! [C] :
( m2_finseq_1(C,k2_zfmisc_1(k7_cqc_lang,k2_cqc_the1))
=> ( ( r2_cqc_the1(A,C)
& r1_tarski(A,B) )
=> r2_cqc_the1(B,C) ) ) ) ) ).
fof(t64_cqc_the1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k7_cqc_lang))
=> ! [C] :
( m2_finseq_1(C,k2_zfmisc_1(k7_cqc_lang,k2_cqc_the1))
=> ( ( r2_cqc_the1(B,C)
& r1_xreal_0(np__1,A)
& r1_xreal_0(A,k3_finseq_1(C)) )
=> r2_hidden(k1_mcart_1(k1_funct_1(C,A)),k1_cqc_the1(B)) ) ) ) ) ).
fof(d6_cqc_the1,axiom,
! [A] :
( m2_finseq_1(A,k2_zfmisc_1(k7_cqc_lang,k2_cqc_the1))
=> ( A != k1_xboole_0
=> k3_cqc_the1(A) = k1_mcart_1(k1_funct_1(A,k3_finseq_1(A))) ) ) ).
fof(t65_cqc_the1,axiom,
$true ).
fof(t66_cqc_the1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> ! [B] :
( m2_finseq_1(B,k2_zfmisc_1(k7_cqc_lang,k2_cqc_the1))
=> ( r2_cqc_the1(A,B)
=> r2_hidden(k3_cqc_the1(B),k1_cqc_the1(A)) ) ) ) ).
fof(t70_cqc_the1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ( r2_hidden(B,k1_cqc_the1(A))
<=> ? [C] :
( m2_finseq_1(C,k2_zfmisc_1(k7_cqc_lang,k2_cqc_the1))
& r2_cqc_the1(A,C)
& k3_cqc_the1(C) = B ) ) ) ) ).
fof(t71_cqc_the1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ~ ( r2_hidden(B,k1_cqc_the1(A))
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k7_cqc_lang))
=> ~ ( r1_tarski(C,A)
& v1_finset_1(C)
& r2_hidden(B,k1_cqc_the1(C)) ) ) ) ) ) ).
fof(d7_cqc_the1,axiom,
$true ).
fof(d8_cqc_the1,axiom,
k4_cqc_the1 = k1_cqc_the1(k1_subset_1(k7_cqc_lang)) ).
fof(t72_cqc_the1,axiom,
$true ).
fof(t73_cqc_the1,axiom,
$true ).
fof(t74_cqc_the1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> ( v1_cqc_the1(A)
=> r1_tarski(k4_cqc_the1,A) ) ) ).
fof(t75_cqc_the1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> r1_tarski(k4_cqc_the1,k1_cqc_the1(A)) ) ).
fof(t76_cqc_the1,axiom,
v1_cqc_the1(k4_cqc_the1) ).
fof(t77_cqc_the1,axiom,
r2_hidden(k9_cqc_lang,k4_cqc_the1) ).
fof(t78_cqc_the1,axiom,
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> r2_hidden(k12_cqc_lang(k12_cqc_lang(k10_cqc_lang(A),A),A),k4_cqc_the1) ) ).
fof(t79_cqc_the1,axiom,
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> r2_hidden(k12_cqc_lang(A,k12_cqc_lang(k10_cqc_lang(A),B)),k4_cqc_the1) ) ) ).
fof(t80_cqc_the1,axiom,
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ! [C] :
( m2_subset_1(C,k8_qc_lang1,k7_cqc_lang)
=> r2_hidden(k12_cqc_lang(k12_cqc_lang(A,B),k12_cqc_lang(k10_cqc_lang(k11_cqc_lang(B,C)),k10_cqc_lang(k11_cqc_lang(A,C)))),k4_cqc_the1) ) ) ) ).
fof(t81_cqc_the1,axiom,
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> r2_hidden(k12_cqc_lang(k11_cqc_lang(A,B),k11_cqc_lang(B,A)),k4_cqc_the1) ) ) ).
fof(t82_cqc_the1,axiom,
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ( ( r2_hidden(A,k4_cqc_the1)
& r2_hidden(k12_cqc_lang(A,B),k4_cqc_the1) )
=> r2_hidden(B,k4_cqc_the1) ) ) ) ).
fof(t83_cqc_the1,axiom,
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ! [B] :
( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1)
=> r2_hidden(k12_cqc_lang(k15_cqc_lang(B,A),A),k4_cqc_the1) ) ) ).
fof(t84_cqc_the1,axiom,
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ! [C] :
( m2_subset_1(C,k1_qc_lang1,k2_qc_lang1)
=> ( r2_hidden(k12_cqc_lang(A,B),k4_cqc_the1)
=> ( r2_hidden(C,k24_qc_lang1(A))
| r2_hidden(k12_cqc_lang(A,k15_cqc_lang(C,B)),k4_cqc_the1) ) ) ) ) ) ).
fof(t85_cqc_the1,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1)
=> ! [C] :
( m2_subset_1(C,k1_qc_lang1,k2_qc_lang1)
=> ( ( r2_hidden(k17_cqc_lang(A,B),k7_cqc_lang)
& r2_hidden(k17_cqc_lang(A,C),k7_cqc_lang)
& r2_hidden(k17_cqc_lang(A,B),k4_cqc_the1) )
=> ( r2_hidden(B,k24_qc_lang1(A))
| r2_hidden(k17_cqc_lang(A,C),k4_cqc_the1) ) ) ) ) ) ).
fof(d9_cqc_the1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> ( r3_cqc_the1(A,B)
<=> r2_hidden(B,k1_cqc_the1(A)) ) ) ) ).
fof(t86_cqc_the1,axiom,
$true ).
fof(t87_cqc_the1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> r3_cqc_the1(A,k9_cqc_lang) ) ).
fof(t88_cqc_the1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> r3_cqc_the1(A,k12_cqc_lang(k12_cqc_lang(k10_cqc_lang(B),B),B)) ) ) ).
fof(t89_cqc_the1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ! [C] :
( m2_subset_1(C,k8_qc_lang1,k7_cqc_lang)
=> r3_cqc_the1(A,k12_cqc_lang(B,k12_cqc_lang(k10_cqc_lang(B),C))) ) ) ) ).
fof(t90_cqc_the1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ! [C] :
( m2_subset_1(C,k8_qc_lang1,k7_cqc_lang)
=> ! [D] :
( m2_subset_1(D,k8_qc_lang1,k7_cqc_lang)
=> r3_cqc_the1(A,k12_cqc_lang(k12_cqc_lang(B,C),k12_cqc_lang(k10_cqc_lang(k11_cqc_lang(C,D)),k10_cqc_lang(k11_cqc_lang(B,D))))) ) ) ) ) ).
fof(t91_cqc_the1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ! [C] :
( m2_subset_1(C,k8_qc_lang1,k7_cqc_lang)
=> r3_cqc_the1(A,k12_cqc_lang(k11_cqc_lang(B,C),k11_cqc_lang(C,B))) ) ) ) ).
fof(t92_cqc_the1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ! [C] :
( m2_subset_1(C,k8_qc_lang1,k7_cqc_lang)
=> ( ( r3_cqc_the1(A,B)
& r3_cqc_the1(A,k12_cqc_lang(B,C)) )
=> r3_cqc_the1(A,C) ) ) ) ) ).
fof(t93_cqc_the1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ! [C] :
( m2_subset_1(C,k1_qc_lang1,k2_qc_lang1)
=> r3_cqc_the1(A,k12_cqc_lang(k15_cqc_lang(C,B),B)) ) ) ) ).
fof(t94_cqc_the1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ! [C] :
( m2_subset_1(C,k8_qc_lang1,k7_cqc_lang)
=> ! [D] :
( m2_subset_1(D,k1_qc_lang1,k2_qc_lang1)
=> ( r3_cqc_the1(A,k12_cqc_lang(B,C))
=> ( r2_hidden(D,k24_qc_lang1(B))
| r3_cqc_the1(A,k12_cqc_lang(B,k15_cqc_lang(D,C))) ) ) ) ) ) ) ).
fof(t95_cqc_the1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> ! [B] :
( m1_subset_1(B,k8_qc_lang1)
=> ! [C] :
( m2_subset_1(C,k1_qc_lang1,k2_qc_lang1)
=> ! [D] :
( m2_subset_1(D,k1_qc_lang1,k2_qc_lang1)
=> ( ( r2_hidden(k17_cqc_lang(B,C),k7_cqc_lang)
& r2_hidden(k17_cqc_lang(B,D),k7_cqc_lang)
& r3_cqc_the1(A,k17_cqc_lang(B,C)) )
=> ( r2_hidden(C,k24_qc_lang1(B))
| r3_cqc_the1(A,k17_cqc_lang(B,D)) ) ) ) ) ) ) ).
fof(d10_cqc_the1,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ( v2_cqc_the1(A)
<=> r3_cqc_the1(k1_subset_1(k7_cqc_lang),A) ) ) ).
fof(d11_cqc_the1,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ( v2_cqc_the1(A)
<=> r2_hidden(A,k4_cqc_the1) ) ) ).
fof(t96_cqc_the1,axiom,
$true ).
fof(t97_cqc_the1,axiom,
$true ).
fof(t98_cqc_the1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ( v2_cqc_the1(B)
=> r3_cqc_the1(A,B) ) ) ) ).
fof(t99_cqc_the1,axiom,
v2_cqc_the1(k9_cqc_lang) ).
fof(t100_cqc_the1,axiom,
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> v2_cqc_the1(k12_cqc_lang(k12_cqc_lang(k10_cqc_lang(A),A),A)) ) ).
fof(t101_cqc_the1,axiom,
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> v2_cqc_the1(k12_cqc_lang(A,k12_cqc_lang(k10_cqc_lang(A),B))) ) ) ).
fof(t102_cqc_the1,axiom,
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ! [C] :
( m2_subset_1(C,k8_qc_lang1,k7_cqc_lang)
=> v2_cqc_the1(k12_cqc_lang(k12_cqc_lang(A,B),k12_cqc_lang(k10_cqc_lang(k11_cqc_lang(B,C)),k10_cqc_lang(k11_cqc_lang(A,C))))) ) ) ) ).
fof(t103_cqc_the1,axiom,
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> v2_cqc_the1(k12_cqc_lang(k11_cqc_lang(A,B),k11_cqc_lang(B,A))) ) ) ).
fof(t104_cqc_the1,axiom,
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ( ( v2_cqc_the1(A)
& v2_cqc_the1(k12_cqc_lang(A,B)) )
=> v2_cqc_the1(B) ) ) ) ).
fof(t105_cqc_the1,axiom,
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ! [B] :
( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1)
=> v2_cqc_the1(k12_cqc_lang(k15_cqc_lang(B,A),A)) ) ) ).
fof(t106_cqc_the1,axiom,
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ! [C] :
( m2_subset_1(C,k1_qc_lang1,k2_qc_lang1)
=> ( v2_cqc_the1(k12_cqc_lang(A,B))
=> ( r2_hidden(C,k24_qc_lang1(A))
| v2_cqc_the1(k12_cqc_lang(A,k15_cqc_lang(C,B))) ) ) ) ) ) ).
fof(t107_cqc_the1,axiom,
! [A] :
( m1_subset_1(A,k8_qc_lang1)
=> ! [B] :
( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1)
=> ! [C] :
( m2_subset_1(C,k1_qc_lang1,k2_qc_lang1)
=> ( ( r2_hidden(k17_cqc_lang(A,B),k7_cqc_lang)
& r2_hidden(k17_cqc_lang(A,C),k7_cqc_lang)
& v2_cqc_the1(k17_cqc_lang(A,B)) )
=> ( r2_hidden(B,k24_qc_lang1(A))
| v2_cqc_the1(k17_cqc_lang(A,C)) ) ) ) ) ) ).
fof(dt_k1_cqc_the1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> m1_subset_1(k1_cqc_the1(A),k1_zfmisc_1(k7_cqc_lang)) ) ).
fof(dt_k2_cqc_the1,axiom,
$true ).
fof(dt_k3_cqc_the1,axiom,
! [A] :
( m1_finseq_1(A,k2_zfmisc_1(k7_cqc_lang,k2_cqc_the1))
=> m2_subset_1(k3_cqc_the1(A),k8_qc_lang1,k7_cqc_lang) ) ).
fof(dt_k4_cqc_the1,axiom,
m1_subset_1(k4_cqc_the1,k1_zfmisc_1(k7_cqc_lang)) ).
fof(t11_cqc_the1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> a_1_0_cqc_the1(A) = k2_xboole_0(a_1_1_cqc_the1(A),k1_tarski(k1_nat_1(A,np__1))) ) ).
fof(t12_cqc_the1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> v1_finset_1(a_1_1_cqc_the1(A)) ) ).
fof(d3_cqc_the1,axiom,
k2_cqc_the1 = a_0_0_cqc_the1 ).
fof(t67_cqc_the1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> r1_tarski(A,a_1_2_cqc_the1(A)) ) ).
fof(t68_cqc_the1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k7_cqc_lang))
=> ( A = a_1_2_cqc_the1(B)
=> v1_cqc_the1(A) ) ) ) ).
fof(t69_cqc_the1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> a_1_2_cqc_the1(A) = k1_cqc_the1(A) ) ).
fof(fraenkel_a_1_0_cqc_the1,axiom,
! [A,B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r2_hidden(A,a_1_0_cqc_the1(B))
<=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& A = C
& r1_xreal_0(C,k1_nat_1(B,np__1)) ) ) ) ).
fof(fraenkel_a_1_1_cqc_the1,axiom,
! [A,B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r2_hidden(A,a_1_1_cqc_the1(B))
<=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& A = C
& r1_xreal_0(C,B) ) ) ) ).
fof(fraenkel_a_0_0_cqc_the1,axiom,
! [A] :
( r2_hidden(A,a_0_0_cqc_the1)
<=> ? [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
& A = B
& r1_xreal_0(B,np__9) ) ) ).
fof(fraenkel_a_1_2_cqc_the1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k7_cqc_lang))
=> ( r2_hidden(A,a_1_2_cqc_the1(B))
<=> ? [C] :
( m2_subset_1(C,k8_qc_lang1,k7_cqc_lang)
& A = C
& ? [D] :
( m2_finseq_1(D,k2_zfmisc_1(k7_cqc_lang,k2_cqc_the1))
& r2_cqc_the1(B,D)
& k3_cqc_the1(D) = C ) ) ) ) ).
%------------------------------------------------------------------------------