SET007 Axioms: SET007+364.ax
%------------------------------------------------------------------------------
% File : SET007+364 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Coherent Space
% 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 : coh_sp [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 133 ( 35 unt; 0 def)
% Number of atoms : 508 ( 108 equ)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 409 ( 34 ~; 2 |; 206 &)
% ( 32 <=>; 135 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 5 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 20 ( 18 usr; 1 prp; 0-3 aty)
% Number of functors : 56 ( 56 usr; 1 con; 0-6 aty)
% Number of variables : 285 ( 258 !; 27 ?)
% SPC :
% Comments : The individual reference can be found in [Mat90] by looking for
% the name provided by [Urb08].
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : These set theory axioms are used in encodings of problems in
% various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(rc1_coh_sp,axiom,
? [A] :
( ~ v1_xboole_0(A)
& v1_classes1(A)
& v1_coh_sp(A) ) ).
fof(fc1_coh_sp,axiom,
! [A] : ~ v1_xboole_0(k3_coh_sp(A)) ).
fof(cc1_coh_sp,axiom,
! [A,B] :
( m1_subset_1(B,k3_coh_sp(A))
=> ( ~ v1_xboole_0(B)
& v1_classes1(B)
& v1_coh_sp(B) ) ) ).
fof(fc2_coh_sp,axiom,
! [A] :
( ~ v1_xboole_0(k4_coh_sp(A))
& v1_fraenkel(k4_coh_sp(A)) ) ).
fof(fc3_coh_sp,axiom,
! [A] : ~ v1_xboole_0(k5_coh_sp(A)) ).
fof(fc4_coh_sp,axiom,
! [A,B] :
( m1_subset_1(B,k5_coh_sp(A))
=> ( v1_relat_1(k2_mcart_1(B))
& v1_funct_1(k2_mcart_1(B)) ) ) ).
fof(fc5_coh_sp,axiom,
! [A] : ~ v1_xboole_0(k15_coh_sp(A)) ).
fof(fc6_coh_sp,axiom,
! [A] : ~ v1_xboole_0(k16_coh_sp(A)) ).
fof(fc7_coh_sp,axiom,
! [A] : ~ v1_xboole_0(k17_coh_sp(A)) ).
fof(fc8_coh_sp,axiom,
! [A] :
( ~ v1_xboole_0(k20_coh_sp(A))
& v1_fraenkel(k20_coh_sp(A)) ) ).
fof(fc9_coh_sp,axiom,
! [A] : ~ v1_xboole_0(k21_coh_sp(A)) ).
fof(fc10_coh_sp,axiom,
! [A,B] :
( m1_subset_1(B,k21_coh_sp(A))
=> ( v1_relat_1(k2_mcart_1(B))
& v1_funct_1(k2_mcart_1(B)) ) ) ).
fof(d1_coh_sp,axiom,
$true ).
fof(d2_coh_sp,axiom,
! [A] :
( v1_coh_sp(A)
<=> ! [B] :
( ( r1_tarski(B,A)
& ! [C,D] :
( ( r2_hidden(C,B)
& r2_hidden(D,B) )
=> r2_hidden(k2_xboole_0(C,D),A) ) )
=> r2_hidden(k3_tarski(B),A) ) ) ).
fof(t1_coh_sp,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes1(A)
& v1_coh_sp(A) )
=> r2_hidden(k1_xboole_0,A) ) ).
fof(t2_coh_sp,axiom,
! [A] :
( ~ v1_xboole_0(k1_zfmisc_1(A))
& v1_classes1(k1_zfmisc_1(A))
& v1_coh_sp(k1_zfmisc_1(A)) ) ).
fof(t3_coh_sp,axiom,
( ~ v1_xboole_0(k1_tarski(k1_xboole_0))
& v1_classes1(k1_tarski(k1_xboole_0))
& v1_coh_sp(k1_tarski(k1_xboole_0)) ) ).
fof(t4_coh_sp,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v1_classes1(B)
& v1_coh_sp(B) )
=> ( r2_hidden(A,k3_tarski(B))
=> r2_hidden(k1_tarski(A),B) ) ) ).
fof(d3_coh_sp,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes1(A)
& v1_coh_sp(A) )
=> ! [B] :
( ( v1_relat_2(B)
& v3_relat_2(B)
& v1_partfun1(B,k3_tarski(A),k3_tarski(A))
& m2_relset_1(B,k3_tarski(A),k3_tarski(A)) )
=> ( B = k1_coh_sp(A)
<=> ! [C,D] :
( r2_hidden(k4_tarski(C,D),B)
<=> ? [E] :
( r2_hidden(E,A)
& r2_hidden(C,E)
& r2_hidden(D,E) ) ) ) ) ) ).
fof(t5_coh_sp,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes1(A)
& v1_coh_sp(A) )
=> ! [B] :
( ( v1_relat_2(B)
& v3_relat_2(B)
& v1_partfun1(B,k3_tarski(A),k3_tarski(A))
& m2_relset_1(B,k3_tarski(A),k3_tarski(A)) )
=> ( B = k1_coh_sp(A)
<=> ! [C,D] :
( r2_hidden(k4_tarski(C,D),B)
<=> r2_hidden(k2_tarski(C,D),A) ) ) ) ) ).
fof(t6_coh_sp,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v1_classes1(B)
& v1_coh_sp(B) )
=> ( r2_hidden(A,B)
<=> ! [C,D] :
( ( r2_hidden(C,A)
& r2_hidden(D,A) )
=> r2_hidden(k2_tarski(C,D),B) ) ) ) ).
fof(t7_coh_sp,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v1_classes1(B)
& v1_coh_sp(B) )
=> ( r2_hidden(A,B)
<=> ! [C,D] :
( ( r2_hidden(C,A)
& r2_hidden(D,A) )
=> r2_hidden(k4_tarski(C,D),k1_coh_sp(B)) ) ) ) ).
fof(t8_coh_sp,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v1_classes1(B)
& v1_coh_sp(B) )
=> ( ! [C,D] :
( ( r2_hidden(C,A)
& r2_hidden(D,A) )
=> r2_hidden(k2_tarski(C,D),B) )
=> r1_tarski(A,k3_tarski(B)) ) ) ).
fof(t9_coh_sp,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes1(A)
& v1_coh_sp(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_classes1(B)
& v1_coh_sp(B) )
=> ( k1_coh_sp(A) = k1_coh_sp(B)
=> A = B ) ) ) ).
fof(t10_coh_sp,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes1(A)
& v1_coh_sp(A) )
=> ( r2_hidden(k3_tarski(A),A)
=> A = k1_zfmisc_1(k3_tarski(A)) ) ) ).
fof(t11_coh_sp,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes1(A)
& v1_coh_sp(A) )
=> ( A = k1_zfmisc_1(k3_tarski(A))
=> k1_coh_sp(A) = k1_eqrel_1(k3_tarski(A)) ) ) ).
fof(d4_coh_sp,axiom,
! [A,B] :
( ( v1_relat_2(B)
& v3_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v1_classes1(C)
& v1_coh_sp(C) )
=> ( C = k2_coh_sp(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> ! [E,F] :
( ( r2_hidden(E,D)
& r2_hidden(F,D) )
=> r2_hidden(k4_tarski(E,F),B) ) ) ) ) ) ).
fof(t12_coh_sp,axiom,
! [A,B] :
( ( v1_relat_2(B)
& v3_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> k1_coh_sp(k2_coh_sp(A,B)) = B ) ).
fof(t13_coh_sp,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes1(A)
& v1_coh_sp(A) )
=> k2_coh_sp(k3_tarski(A),k1_coh_sp(A)) = A ) ).
fof(t14_coh_sp,axiom,
! [A,B,C] :
( ( v1_relat_2(C)
& v3_relat_2(C)
& v1_partfun1(C,A,A)
& m2_relset_1(C,A,A) )
=> ( r2_hidden(B,k2_coh_sp(A,C))
<=> m1_toler_1(B,A,C) ) ) ).
fof(t15_coh_sp,axiom,
! [A,B] :
( ( v1_relat_2(B)
& v3_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> k2_coh_sp(A,B) = k3_toler_1(A,B) ) ).
fof(t16_coh_sp,axiom,
! [A,B,C,D] :
( m1_subset_1(D,k3_coh_sp(A))
=> ( r2_hidden(k2_tarski(B,C),D)
=> ( r2_hidden(B,k3_tarski(D))
& r2_hidden(C,k3_tarski(D)) ) ) ) ).
fof(d6_coh_sp,axiom,
$true ).
fof(t17_coh_sp,axiom,
! [A,B] :
( r2_hidden(A,k4_coh_sp(B))
<=> ? [C] :
( m1_subset_1(C,k3_coh_sp(B))
& ? [D] :
( m1_subset_1(D,k3_coh_sp(B))
& ( k3_tarski(D) = k1_xboole_0
=> k3_tarski(C) = k1_xboole_0 )
& v1_funct_1(A)
& v1_funct_2(A,k3_tarski(C),k3_tarski(D))
& m2_relset_1(A,k3_tarski(C),k3_tarski(D)) ) ) ) ).
fof(t18_coh_sp,axiom,
! [A,B] :
( m1_subset_1(B,k5_coh_sp(A))
=> ? [C] :
( m1_subset_1(C,k4_coh_sp(A))
& ? [D] :
( m1_subset_1(D,k3_coh_sp(A))
& ? [E] :
( m1_subset_1(E,k3_coh_sp(A))
& B = k4_tarski(k4_tarski(D,E),C)
& ( k3_tarski(E) = k1_xboole_0
=> k3_tarski(D) = k1_xboole_0 )
& v1_funct_1(C)
& v1_funct_2(C,k3_tarski(D),k3_tarski(E))
& m2_relset_1(C,k3_tarski(D),k3_tarski(E))
& ! [F,G] :
( r2_hidden(k2_tarski(F,G),D)
=> r2_hidden(k2_tarski(k1_funct_1(C,F),k1_funct_1(C,G)),E) ) ) ) ) ) ).
fof(t19_coh_sp,axiom,
! [A,B] :
( m1_subset_1(B,k3_coh_sp(A))
=> ! [C] :
( m1_subset_1(C,k3_coh_sp(A))
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k3_tarski(B),k3_tarski(C))
& m2_relset_1(D,k3_tarski(B),k3_tarski(C)) )
=> ( ! [E,F] :
( r2_hidden(k2_tarski(E,F),B)
=> r2_hidden(k2_tarski(k1_funct_1(D,E),k1_funct_1(D,F)),C) )
=> ( ( k3_tarski(C) = k1_xboole_0
& k3_tarski(B) != k1_xboole_0 )
| r2_hidden(k4_tarski(k4_tarski(B,C),D),k5_coh_sp(A)) ) ) ) ) ) ).
fof(d9_coh_sp,axiom,
$true ).
fof(d10_coh_sp,axiom,
! [A,B] :
( m1_subset_1(B,k5_coh_sp(A))
=> k6_coh_sp(A,B) = k1_mcart_1(k1_mcart_1(B)) ) ).
fof(d11_coh_sp,axiom,
! [A,B] :
( m1_subset_1(B,k5_coh_sp(A))
=> k7_coh_sp(A,B) = k2_mcart_1(k1_mcart_1(B)) ) ).
fof(t20_coh_sp,axiom,
! [A,B] :
( m1_subset_1(B,k5_coh_sp(A))
=> B = k4_tarski(k4_tarski(k6_coh_sp(A,B),k7_coh_sp(A,B)),k2_mcart_1(B)) ) ).
fof(d12_coh_sp,axiom,
! [A,B] :
( m1_subset_1(B,k3_coh_sp(A))
=> k8_coh_sp(A,B) = k4_tarski(k4_tarski(B,B),k6_partfun1(k3_tarski(B))) ) ).
fof(t21_coh_sp,axiom,
! [A,B] :
( m1_subset_1(B,k5_coh_sp(A))
=> ( ~ ( k3_tarski(k7_coh_sp(A,B)) = k1_xboole_0
& k3_tarski(k6_coh_sp(A,B)) != k1_xboole_0 )
& v1_funct_1(k2_mcart_1(B))
& v1_funct_2(k2_mcart_1(B),k3_tarski(k6_coh_sp(A,B)),k3_tarski(k7_coh_sp(A,B)))
& m2_relset_1(k2_mcart_1(B),k3_tarski(k6_coh_sp(A,B)),k3_tarski(k7_coh_sp(A,B)))
& ! [C,D] :
( r2_hidden(k2_tarski(C,D),k6_coh_sp(A,B))
=> r2_hidden(k2_tarski(k1_funct_1(k2_mcart_1(B),C),k1_funct_1(k2_mcart_1(B),D)),k7_coh_sp(A,B)) ) ) ) ).
fof(d13_coh_sp,axiom,
! [A,B] :
( m1_subset_1(B,k5_coh_sp(A))
=> ! [C] :
( m1_subset_1(C,k5_coh_sp(A))
=> ( k7_coh_sp(A,B) = k6_coh_sp(A,C)
=> k9_coh_sp(A,B,C) = k4_tarski(k4_tarski(k6_coh_sp(A,B),k7_coh_sp(A,C)),k5_relat_1(k2_mcart_1(B),k2_mcart_1(C))) ) ) ) ).
fof(t22_coh_sp,axiom,
! [A,B] :
( m1_subset_1(B,k5_coh_sp(A))
=> ! [C] :
( m1_subset_1(C,k5_coh_sp(A))
=> ( k6_coh_sp(A,B) = k7_coh_sp(A,C)
=> ( k2_mcart_1(k9_coh_sp(A,C,B)) = k5_relat_1(k2_mcart_1(C),k2_mcart_1(B))
& k6_coh_sp(A,k9_coh_sp(A,C,B)) = k6_coh_sp(A,C)
& k7_coh_sp(A,k9_coh_sp(A,C,B)) = k7_coh_sp(A,B) ) ) ) ) ).
fof(t23_coh_sp,axiom,
! [A,B] :
( m1_subset_1(B,k5_coh_sp(A))
=> ! [C] :
( m1_subset_1(C,k5_coh_sp(A))
=> ! [D] :
( m1_subset_1(D,k5_coh_sp(A))
=> ( ( k6_coh_sp(A,B) = k7_coh_sp(A,C)
& k6_coh_sp(A,D) = k7_coh_sp(A,B) )
=> k9_coh_sp(A,k9_coh_sp(A,C,B),D) = k9_coh_sp(A,C,k9_coh_sp(A,B,D)) ) ) ) ) ).
fof(t24_coh_sp,axiom,
! [A,B] :
( m1_subset_1(B,k3_coh_sp(A))
=> ( k2_mcart_1(k8_coh_sp(A,B)) = k6_partfun1(k3_tarski(B))
& k6_coh_sp(A,k8_coh_sp(A,B)) = B
& k7_coh_sp(A,k8_coh_sp(A,B)) = B ) ) ).
fof(t25_coh_sp,axiom,
! [A,B] :
( m1_subset_1(B,k5_coh_sp(A))
=> ( k9_coh_sp(A,k8_coh_sp(A,k6_coh_sp(A,B)),B) = B
& k9_coh_sp(A,B,k8_coh_sp(A,k7_coh_sp(A,B))) = B ) ) ).
fof(d14_coh_sp,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_coh_sp(A),k3_coh_sp(A))
& m2_relset_1(B,k5_coh_sp(A),k3_coh_sp(A)) )
=> ( B = k10_coh_sp(A)
<=> ! [C] :
( m1_subset_1(C,k5_coh_sp(A))
=> k8_funct_2(k5_coh_sp(A),k3_coh_sp(A),B,C) = k6_coh_sp(A,C) ) ) ) ).
fof(d15_coh_sp,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_coh_sp(A),k3_coh_sp(A))
& m2_relset_1(B,k5_coh_sp(A),k3_coh_sp(A)) )
=> ( B = k11_coh_sp(A)
<=> ! [C] :
( m1_subset_1(C,k5_coh_sp(A))
=> k8_funct_2(k5_coh_sp(A),k3_coh_sp(A),B,C) = k7_coh_sp(A,C) ) ) ) ).
fof(d16_coh_sp,axiom,
! [A,B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k2_zfmisc_1(k5_coh_sp(A),k5_coh_sp(A)),k5_coh_sp(A)) )
=> ( B = k12_coh_sp(A)
<=> ( ! [C] :
( m1_subset_1(C,k5_coh_sp(A))
=> ! [D] :
( m1_subset_1(D,k5_coh_sp(A))
=> ( r2_hidden(k4_tarski(C,D),k4_relset_1(k2_zfmisc_1(k5_coh_sp(A),k5_coh_sp(A)),k5_coh_sp(A),B))
<=> k6_coh_sp(A,C) = k7_coh_sp(A,D) ) ) )
& ! [C] :
( m1_subset_1(C,k5_coh_sp(A))
=> ! [D] :
( m1_subset_1(D,k5_coh_sp(A))
=> ( k6_coh_sp(A,C) = k7_coh_sp(A,D)
=> k1_funct_1(B,k4_tarski(C,D)) = k9_coh_sp(A,D,C) ) ) ) ) ) ) ).
fof(d17_coh_sp,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k3_coh_sp(A),k5_coh_sp(A))
& m2_relset_1(B,k3_coh_sp(A),k5_coh_sp(A)) )
=> ( B = k13_coh_sp(A)
<=> ! [C] :
( m1_subset_1(C,k3_coh_sp(A))
=> k8_funct_2(k3_coh_sp(A),k5_coh_sp(A),B,C) = k8_coh_sp(A,C) ) ) ) ).
fof(t26_coh_sp,axiom,
! [A] :
( v2_cat_1(g1_cat_1(k3_coh_sp(A),k5_coh_sp(A),k10_coh_sp(A),k11_coh_sp(A),k12_coh_sp(A),k13_coh_sp(A)))
& l1_cat_1(g1_cat_1(k3_coh_sp(A),k5_coh_sp(A),k10_coh_sp(A),k11_coh_sp(A),k12_coh_sp(A),k13_coh_sp(A))) ) ).
fof(d18_coh_sp,axiom,
! [A] : k14_coh_sp(A) = g1_cat_1(k3_coh_sp(A),k5_coh_sp(A),k10_coh_sp(A),k11_coh_sp(A),k12_coh_sp(A),k13_coh_sp(A)) ).
fof(d19_coh_sp,axiom,
! [A,B] :
( B = k15_coh_sp(A)
<=> ! [C] :
( r2_hidden(C,B)
<=> ( v1_relat_2(C)
& v3_relat_2(C)
& v1_partfun1(C,A,A)
& m2_relset_1(C,A,A) ) ) ) ).
fof(t27_coh_sp,axiom,
! [A,B] :
( r2_hidden(A,k16_coh_sp(B))
<=> ? [C] :
( r1_tarski(C,B)
& v1_relat_2(A)
& v3_relat_2(A)
& v1_partfun1(A,C,C)
& m2_relset_1(A,C,C) ) ) ).
fof(t28_coh_sp,axiom,
! [A] : r2_hidden(k1_eqrel_1(A),k15_coh_sp(A)) ).
fof(t29_coh_sp,axiom,
$true ).
fof(t30_coh_sp,axiom,
! [A] : r2_hidden(k1_xboole_0,k16_coh_sp(A)) ).
fof(t31_coh_sp,axiom,
! [A,B] :
( r1_tarski(A,B)
=> r2_hidden(k1_eqrel_1(A),k16_coh_sp(B)) ) ).
fof(t32_coh_sp,axiom,
! [A,B] :
( r1_tarski(A,B)
=> r2_hidden(k6_partfun1(A),k16_coh_sp(B)) ) ).
fof(t33_coh_sp,axiom,
! [A] : r2_hidden(k1_eqrel_1(A),k16_coh_sp(A)) ).
fof(t34_coh_sp,axiom,
! [A] : r2_hidden(k6_partfun1(A),k16_coh_sp(A)) ).
fof(t35_coh_sp,axiom,
! [A] : r2_hidden(k4_tarski(k1_xboole_0,k1_xboole_0),k17_coh_sp(A)) ).
fof(t36_coh_sp,axiom,
! [A,B] :
( r1_tarski(A,B)
=> r2_hidden(k4_tarski(k6_partfun1(A),A),k17_coh_sp(B)) ) ).
fof(t37_coh_sp,axiom,
! [A,B] :
( r1_tarski(A,B)
=> r2_hidden(k4_tarski(k1_eqrel_1(A),A),k17_coh_sp(B)) ) ).
fof(t38_coh_sp,axiom,
! [A] : r2_hidden(k4_tarski(k6_partfun1(A),A),k17_coh_sp(A)) ).
fof(t39_coh_sp,axiom,
! [A] : r2_hidden(k4_tarski(k1_eqrel_1(A),A),k17_coh_sp(A)) ).
fof(t40_coh_sp,axiom,
! [A,B] :
( r2_hidden(A,k20_coh_sp(B))
<=> ? [C] :
( m1_subset_1(C,k17_coh_sp(B))
& ? [D] :
( m1_subset_1(D,k17_coh_sp(B))
& ( k18_coh_sp(B,D) = k1_xboole_0
=> k18_coh_sp(B,C) = k1_xboole_0 )
& v1_funct_1(A)
& v1_funct_2(A,k18_coh_sp(B,C),k18_coh_sp(B,D))
& m2_relset_1(A,k18_coh_sp(B,C),k18_coh_sp(B,D)) ) ) ) ).
fof(t41_coh_sp,axiom,
! [A,B] :
( m1_subset_1(B,k21_coh_sp(A))
=> ? [C] :
( m1_subset_1(C,k20_coh_sp(A))
& ? [D] :
( m1_subset_1(D,k17_coh_sp(A))
& ? [E] :
( m1_subset_1(E,k17_coh_sp(A))
& B = k4_tarski(k4_tarski(D,E),C)
& ( k18_coh_sp(A,E) = k1_xboole_0
=> k18_coh_sp(A,D) = k1_xboole_0 )
& v1_funct_1(C)
& v1_funct_2(C,k18_coh_sp(A,D),k18_coh_sp(A,E))
& m2_relset_1(C,k18_coh_sp(A,D),k18_coh_sp(A,E))
& ! [F,G] :
( r2_hidden(k4_tarski(F,G),k19_coh_sp(A,D))
=> r2_hidden(k4_tarski(k1_funct_1(C,F),k1_funct_1(C,G)),k19_coh_sp(A,E)) ) ) ) ) ) ).
fof(t42_coh_sp,axiom,
! [A,B] :
( m1_subset_1(B,k17_coh_sp(A))
=> ! [C] :
( m1_subset_1(C,k17_coh_sp(A))
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k18_coh_sp(A,B),k18_coh_sp(A,C))
& m2_relset_1(D,k18_coh_sp(A,B),k18_coh_sp(A,C)) )
=> ( ! [E,F] :
( r2_hidden(k4_tarski(E,F),k19_coh_sp(A,B))
=> r2_hidden(k4_tarski(k1_funct_1(D,E),k1_funct_1(D,F)),k19_coh_sp(A,C)) )
=> ( ( k18_coh_sp(A,C) = k1_xboole_0
& k18_coh_sp(A,B) != k1_xboole_0 )
| r2_hidden(k4_tarski(k4_tarski(B,C),D),k21_coh_sp(A)) ) ) ) ) ) ).
fof(d24_coh_sp,axiom,
$true ).
fof(d25_coh_sp,axiom,
! [A,B] :
( m1_subset_1(B,k21_coh_sp(A))
=> k22_coh_sp(A,B) = k1_mcart_1(k1_mcart_1(B)) ) ).
fof(d26_coh_sp,axiom,
! [A,B] :
( m1_subset_1(B,k21_coh_sp(A))
=> k23_coh_sp(A,B) = k2_mcart_1(k1_mcart_1(B)) ) ).
fof(t43_coh_sp,axiom,
! [A,B] :
( m1_subset_1(B,k21_coh_sp(A))
=> B = k4_tarski(k4_tarski(k22_coh_sp(A,B),k23_coh_sp(A,B)),k2_mcart_1(B)) ) ).
fof(d27_coh_sp,axiom,
! [A,B] :
( m1_subset_1(B,k17_coh_sp(A))
=> k24_coh_sp(A,B) = k4_tarski(k4_tarski(B,B),k6_partfun1(k18_coh_sp(A,B))) ) ).
fof(t44_coh_sp,axiom,
! [A,B] :
( m1_subset_1(B,k21_coh_sp(A))
=> ( ~ ( k18_coh_sp(A,k23_coh_sp(A,B)) = k1_xboole_0
& k18_coh_sp(A,k22_coh_sp(A,B)) != k1_xboole_0 )
& v1_funct_1(k2_mcart_1(B))
& v1_funct_2(k2_mcart_1(B),k18_coh_sp(A,k22_coh_sp(A,B)),k18_coh_sp(A,k23_coh_sp(A,B)))
& m2_relset_1(k2_mcart_1(B),k18_coh_sp(A,k22_coh_sp(A,B)),k18_coh_sp(A,k23_coh_sp(A,B)))
& ! [C,D] :
( r2_hidden(k4_tarski(C,D),k19_coh_sp(A,k22_coh_sp(A,B)))
=> r2_hidden(k4_tarski(k1_funct_1(k2_mcart_1(B),C),k1_funct_1(k2_mcart_1(B),D)),k19_coh_sp(A,k23_coh_sp(A,B))) ) ) ) ).
fof(d28_coh_sp,axiom,
! [A,B] :
( m1_subset_1(B,k21_coh_sp(A))
=> ! [C] :
( m1_subset_1(C,k21_coh_sp(A))
=> ( k23_coh_sp(A,B) = k22_coh_sp(A,C)
=> k25_coh_sp(A,B,C) = k4_tarski(k4_tarski(k22_coh_sp(A,B),k23_coh_sp(A,C)),k5_relat_1(k2_mcart_1(B),k2_mcart_1(C))) ) ) ) ).
fof(t45_coh_sp,axiom,
! [A,B] :
( m1_subset_1(B,k21_coh_sp(A))
=> ! [C] :
( m1_subset_1(C,k21_coh_sp(A))
=> ( k22_coh_sp(A,B) = k23_coh_sp(A,C)
=> ( k2_mcart_1(k25_coh_sp(A,C,B)) = k5_relat_1(k2_mcart_1(C),k2_mcart_1(B))
& k22_coh_sp(A,k25_coh_sp(A,C,B)) = k22_coh_sp(A,C)
& k23_coh_sp(A,k25_coh_sp(A,C,B)) = k23_coh_sp(A,B) ) ) ) ) ).
fof(t46_coh_sp,axiom,
! [A,B] :
( m1_subset_1(B,k21_coh_sp(A))
=> ! [C] :
( m1_subset_1(C,k21_coh_sp(A))
=> ! [D] :
( m1_subset_1(D,k21_coh_sp(A))
=> ( ( k22_coh_sp(A,B) = k23_coh_sp(A,C)
& k22_coh_sp(A,D) = k23_coh_sp(A,B) )
=> k25_coh_sp(A,k25_coh_sp(A,C,B),D) = k25_coh_sp(A,C,k25_coh_sp(A,B,D)) ) ) ) ) ).
fof(t47_coh_sp,axiom,
! [A,B] :
( m1_subset_1(B,k17_coh_sp(A))
=> ( k2_mcart_1(k24_coh_sp(A,B)) = k6_partfun1(k18_coh_sp(A,B))
& k22_coh_sp(A,k24_coh_sp(A,B)) = B
& k23_coh_sp(A,k24_coh_sp(A,B)) = B ) ) ).
fof(t48_coh_sp,axiom,
! [A,B] :
( m1_subset_1(B,k21_coh_sp(A))
=> ( k25_coh_sp(A,k24_coh_sp(A,k22_coh_sp(A,B)),B) = B
& k25_coh_sp(A,B,k24_coh_sp(A,k23_coh_sp(A,B))) = B ) ) ).
fof(d29_coh_sp,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k21_coh_sp(A),k17_coh_sp(A))
& m2_relset_1(B,k21_coh_sp(A),k17_coh_sp(A)) )
=> ( B = k26_coh_sp(A)
<=> ! [C] :
( m1_subset_1(C,k21_coh_sp(A))
=> k8_funct_2(k21_coh_sp(A),k17_coh_sp(A),B,C) = k22_coh_sp(A,C) ) ) ) ).
fof(d30_coh_sp,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k21_coh_sp(A),k17_coh_sp(A))
& m2_relset_1(B,k21_coh_sp(A),k17_coh_sp(A)) )
=> ( B = k27_coh_sp(A)
<=> ! [C] :
( m1_subset_1(C,k21_coh_sp(A))
=> k8_funct_2(k21_coh_sp(A),k17_coh_sp(A),B,C) = k23_coh_sp(A,C) ) ) ) ).
fof(d31_coh_sp,axiom,
! [A,B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k2_zfmisc_1(k21_coh_sp(A),k21_coh_sp(A)),k21_coh_sp(A)) )
=> ( B = k28_coh_sp(A)
<=> ( ! [C] :
( m1_subset_1(C,k21_coh_sp(A))
=> ! [D] :
( m1_subset_1(D,k21_coh_sp(A))
=> ( r2_hidden(k4_tarski(C,D),k4_relset_1(k2_zfmisc_1(k21_coh_sp(A),k21_coh_sp(A)),k21_coh_sp(A),B))
<=> k22_coh_sp(A,C) = k23_coh_sp(A,D) ) ) )
& ! [C] :
( m1_subset_1(C,k21_coh_sp(A))
=> ! [D] :
( m1_subset_1(D,k21_coh_sp(A))
=> ( k22_coh_sp(A,C) = k23_coh_sp(A,D)
=> k1_funct_1(B,k4_tarski(C,D)) = k25_coh_sp(A,D,C) ) ) ) ) ) ) ).
fof(d32_coh_sp,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k17_coh_sp(A),k21_coh_sp(A))
& m2_relset_1(B,k17_coh_sp(A),k21_coh_sp(A)) )
=> ( B = k29_coh_sp(A)
<=> ! [C] :
( m1_subset_1(C,k17_coh_sp(A))
=> k8_funct_2(k17_coh_sp(A),k21_coh_sp(A),B,C) = k24_coh_sp(A,C) ) ) ) ).
fof(t49_coh_sp,axiom,
! [A] :
( v2_cat_1(g1_cat_1(k17_coh_sp(A),k21_coh_sp(A),k26_coh_sp(A),k27_coh_sp(A),k28_coh_sp(A),k29_coh_sp(A)))
& l1_cat_1(g1_cat_1(k17_coh_sp(A),k21_coh_sp(A),k26_coh_sp(A),k27_coh_sp(A),k28_coh_sp(A),k29_coh_sp(A))) ) ).
fof(d33_coh_sp,axiom,
! [A] : k30_coh_sp(A) = g1_cat_1(k17_coh_sp(A),k21_coh_sp(A),k26_coh_sp(A),k27_coh_sp(A),k28_coh_sp(A),k29_coh_sp(A)) ).
fof(dt_k1_coh_sp,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes1(A)
& v1_coh_sp(A) )
=> ( v1_relat_2(k1_coh_sp(A))
& v3_relat_2(k1_coh_sp(A))
& v1_partfun1(k1_coh_sp(A),k3_tarski(A),k3_tarski(A))
& m2_relset_1(k1_coh_sp(A),k3_tarski(A),k3_tarski(A)) ) ) ).
fof(dt_k2_coh_sp,axiom,
! [A,B] :
( ( v1_relat_2(B)
& v3_relat_2(B)
& v1_partfun1(B,A,A)
& m1_relset_1(B,A,A) )
=> ( ~ v1_xboole_0(k2_coh_sp(A,B))
& v1_classes1(k2_coh_sp(A,B))
& v1_coh_sp(k2_coh_sp(A,B)) ) ) ).
fof(dt_k3_coh_sp,axiom,
$true ).
fof(dt_k4_coh_sp,axiom,
$true ).
fof(dt_k5_coh_sp,axiom,
$true ).
fof(dt_k6_coh_sp,axiom,
! [A,B] :
( m1_subset_1(B,k5_coh_sp(A))
=> m1_subset_1(k6_coh_sp(A,B),k3_coh_sp(A)) ) ).
fof(dt_k7_coh_sp,axiom,
! [A,B] :
( m1_subset_1(B,k5_coh_sp(A))
=> m1_subset_1(k7_coh_sp(A,B),k3_coh_sp(A)) ) ).
fof(dt_k8_coh_sp,axiom,
! [A,B] :
( m1_subset_1(B,k3_coh_sp(A))
=> m1_subset_1(k8_coh_sp(A,B),k5_coh_sp(A)) ) ).
fof(dt_k9_coh_sp,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k5_coh_sp(A))
& m1_subset_1(C,k5_coh_sp(A)) )
=> m1_subset_1(k9_coh_sp(A,B,C),k5_coh_sp(A)) ) ).
fof(dt_k10_coh_sp,axiom,
! [A] :
( v1_funct_1(k10_coh_sp(A))
& v1_funct_2(k10_coh_sp(A),k5_coh_sp(A),k3_coh_sp(A))
& m2_relset_1(k10_coh_sp(A),k5_coh_sp(A),k3_coh_sp(A)) ) ).
fof(dt_k11_coh_sp,axiom,
! [A] :
( v1_funct_1(k11_coh_sp(A))
& v1_funct_2(k11_coh_sp(A),k5_coh_sp(A),k3_coh_sp(A))
& m2_relset_1(k11_coh_sp(A),k5_coh_sp(A),k3_coh_sp(A)) ) ).
fof(dt_k12_coh_sp,axiom,
! [A] :
( v1_funct_1(k12_coh_sp(A))
& m2_relset_1(k12_coh_sp(A),k2_zfmisc_1(k5_coh_sp(A),k5_coh_sp(A)),k5_coh_sp(A)) ) ).
fof(dt_k13_coh_sp,axiom,
! [A] :
( v1_funct_1(k13_coh_sp(A))
& v1_funct_2(k13_coh_sp(A),k3_coh_sp(A),k5_coh_sp(A))
& m2_relset_1(k13_coh_sp(A),k3_coh_sp(A),k5_coh_sp(A)) ) ).
fof(dt_k14_coh_sp,axiom,
! [A] :
( v2_cat_1(k14_coh_sp(A))
& l1_cat_1(k14_coh_sp(A)) ) ).
fof(dt_k15_coh_sp,axiom,
$true ).
fof(dt_k16_coh_sp,axiom,
$true ).
fof(dt_k17_coh_sp,axiom,
$true ).
fof(dt_k18_coh_sp,axiom,
! [A,B] :
( m1_subset_1(B,k17_coh_sp(A))
=> m1_subset_1(k18_coh_sp(A,B),k1_zfmisc_1(A)) ) ).
fof(redefinition_k18_coh_sp,axiom,
! [A,B] :
( m1_subset_1(B,k17_coh_sp(A))
=> k18_coh_sp(A,B) = k2_mcart_1(B) ) ).
fof(dt_k19_coh_sp,axiom,
! [A,B] :
( m1_subset_1(B,k17_coh_sp(A))
=> ( v1_relat_2(k19_coh_sp(A,B))
& v3_relat_2(k19_coh_sp(A,B))
& v1_partfun1(k19_coh_sp(A,B),k2_mcart_1(B),k2_mcart_1(B))
& m2_relset_1(k19_coh_sp(A,B),k2_mcart_1(B),k2_mcart_1(B)) ) ) ).
fof(redefinition_k19_coh_sp,axiom,
! [A,B] :
( m1_subset_1(B,k17_coh_sp(A))
=> k19_coh_sp(A,B) = k1_mcart_1(B) ) ).
fof(dt_k20_coh_sp,axiom,
$true ).
fof(dt_k21_coh_sp,axiom,
$true ).
fof(dt_k22_coh_sp,axiom,
! [A,B] :
( m1_subset_1(B,k21_coh_sp(A))
=> m1_subset_1(k22_coh_sp(A,B),k17_coh_sp(A)) ) ).
fof(dt_k23_coh_sp,axiom,
! [A,B] :
( m1_subset_1(B,k21_coh_sp(A))
=> m1_subset_1(k23_coh_sp(A,B),k17_coh_sp(A)) ) ).
fof(dt_k24_coh_sp,axiom,
! [A,B] :
( m1_subset_1(B,k17_coh_sp(A))
=> m1_subset_1(k24_coh_sp(A,B),k21_coh_sp(A)) ) ).
fof(dt_k25_coh_sp,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k21_coh_sp(A))
& m1_subset_1(C,k21_coh_sp(A)) )
=> m1_subset_1(k25_coh_sp(A,B,C),k21_coh_sp(A)) ) ).
fof(dt_k26_coh_sp,axiom,
! [A] :
( v1_funct_1(k26_coh_sp(A))
& v1_funct_2(k26_coh_sp(A),k21_coh_sp(A),k17_coh_sp(A))
& m2_relset_1(k26_coh_sp(A),k21_coh_sp(A),k17_coh_sp(A)) ) ).
fof(dt_k27_coh_sp,axiom,
! [A] :
( v1_funct_1(k27_coh_sp(A))
& v1_funct_2(k27_coh_sp(A),k21_coh_sp(A),k17_coh_sp(A))
& m2_relset_1(k27_coh_sp(A),k21_coh_sp(A),k17_coh_sp(A)) ) ).
fof(dt_k28_coh_sp,axiom,
! [A] :
( v1_funct_1(k28_coh_sp(A))
& m2_relset_1(k28_coh_sp(A),k2_zfmisc_1(k21_coh_sp(A),k21_coh_sp(A)),k21_coh_sp(A)) ) ).
fof(dt_k29_coh_sp,axiom,
! [A] :
( v1_funct_1(k29_coh_sp(A))
& v1_funct_2(k29_coh_sp(A),k17_coh_sp(A),k21_coh_sp(A))
& m2_relset_1(k29_coh_sp(A),k17_coh_sp(A),k21_coh_sp(A)) ) ).
fof(dt_k30_coh_sp,axiom,
! [A] :
( v2_cat_1(k30_coh_sp(A))
& l1_cat_1(k30_coh_sp(A)) ) ).
fof(d5_coh_sp,axiom,
! [A] : k3_coh_sp(A) = a_1_0_coh_sp(A) ).
fof(d7_coh_sp,axiom,
! [A] : k4_coh_sp(A) = k3_tarski(a_1_1_coh_sp(A)) ).
fof(d8_coh_sp,axiom,
! [A] : k5_coh_sp(A) = a_1_2_coh_sp(A) ).
fof(d20_coh_sp,axiom,
! [A] : k16_coh_sp(A) = k3_tarski(a_1_3_coh_sp(A)) ).
fof(d21_coh_sp,axiom,
! [A] : k17_coh_sp(A) = a_1_4_coh_sp(A) ).
fof(d22_coh_sp,axiom,
! [A] : k20_coh_sp(A) = k3_tarski(a_1_5_coh_sp(A)) ).
fof(d23_coh_sp,axiom,
! [A] : k21_coh_sp(A) = a_1_6_coh_sp(A) ).
fof(fraenkel_a_1_0_coh_sp,axiom,
! [A,B] :
( r2_hidden(A,a_1_0_coh_sp(B))
<=> ? [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(B)))
& A = C
& ~ v1_xboole_0(C)
& v1_classes1(C)
& v1_coh_sp(C) ) ) ).
fof(fraenkel_a_1_1_coh_sp,axiom,
! [A,B] :
( r2_hidden(A,a_1_1_coh_sp(B))
<=> ? [C,D] :
( m1_subset_1(C,k3_coh_sp(B))
& m1_subset_1(D,k3_coh_sp(B))
& A = k1_funct_2(k3_tarski(C),k3_tarski(D)) ) ) ).
fof(fraenkel_a_1_2_coh_sp,axiom,
! [A,B] :
( r2_hidden(A,a_1_2_coh_sp(B))
<=> ? [C,D,E] :
( m1_subset_1(C,k3_coh_sp(B))
& m1_subset_1(D,k3_coh_sp(B))
& m1_subset_1(E,k4_coh_sp(B))
& A = k4_tarski(k4_tarski(C,D),E)
& ( k3_tarski(D) = k1_xboole_0
=> k3_tarski(C) = k1_xboole_0 )
& v1_funct_1(E)
& v1_funct_2(E,k3_tarski(C),k3_tarski(D))
& m2_relset_1(E,k3_tarski(C),k3_tarski(D))
& ! [F,G] :
( r2_hidden(k2_tarski(F,G),C)
=> r2_hidden(k2_tarski(k1_funct_1(E,F),k1_funct_1(E,G)),D) ) ) ) ).
fof(fraenkel_a_1_3_coh_sp,axiom,
! [A,B] :
( r2_hidden(A,a_1_3_coh_sp(B))
<=> ? [C] :
( m1_subset_1(C,k1_zfmisc_1(B))
& A = k15_coh_sp(C) ) ) ).
fof(fraenkel_a_1_4_coh_sp,axiom,
! [A,B] :
( r2_hidden(A,a_1_4_coh_sp(B))
<=> ? [C,D] :
( m1_subset_1(C,k16_coh_sp(B))
& m1_subset_1(D,k1_zfmisc_1(B))
& A = k4_tarski(C,D)
& v1_relat_2(C)
& v3_relat_2(C)
& v1_partfun1(C,D,D)
& m2_relset_1(C,D,D) ) ) ).
fof(fraenkel_a_1_5_coh_sp,axiom,
! [A,B] :
( r2_hidden(A,a_1_5_coh_sp(B))
<=> ? [C,D] :
( m1_subset_1(C,k17_coh_sp(B))
& m1_subset_1(D,k17_coh_sp(B))
& A = k1_funct_2(k18_coh_sp(B,C),k18_coh_sp(B,D)) ) ) ).
fof(fraenkel_a_1_6_coh_sp,axiom,
! [A,B] :
( r2_hidden(A,a_1_6_coh_sp(B))
<=> ? [C,D,E] :
( m1_subset_1(C,k17_coh_sp(B))
& m1_subset_1(D,k17_coh_sp(B))
& m1_subset_1(E,k20_coh_sp(B))
& A = k4_tarski(k4_tarski(C,D),E)
& ( k18_coh_sp(B,D) = k1_xboole_0
=> k18_coh_sp(B,C) = k1_xboole_0 )
& v1_funct_1(E)
& v1_funct_2(E,k18_coh_sp(B,C),k18_coh_sp(B,D))
& m2_relset_1(E,k18_coh_sp(B,C),k18_coh_sp(B,D))
& ! [F,G] :
( r2_hidden(k4_tarski(F,G),k19_coh_sp(B,C))
=> r2_hidden(k4_tarski(k1_funct_1(E,F),k1_funct_1(E,G)),k19_coh_sp(B,D)) ) ) ) ).
%------------------------------------------------------------------------------