SET007 Axioms: SET007+455.ax
%------------------------------------------------------------------------------
% File : SET007+455 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : On the Closure Operator and the Closure System of Many Sorted Sets
% 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 : closure2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 140 ( 4 unt; 0 def)
% Number of atoms : 846 ( 52 equ)
% Maximal formula atoms : 14 ( 6 avg)
% Number of connectives : 731 ( 25 ~; 0 |; 360 &)
% ( 30 <=>; 316 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 8 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 46 ( 44 usr; 1 prp; 0-4 aty)
% Number of functors : 48 ( 48 usr; 1 con; 0-4 aty)
% Number of variables : 435 ( 418 !; 17 ?)
% 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(cc1_closure2,axiom,
! [A] :
( v1_xboole_0(A)
=> v1_fraenkel(A) ) ).
fof(rc1_closure2,axiom,
? [A] :
( v1_xboole_0(A)
& v1_relat_1(A)
& v1_funct_1(A)
& v2_funct_1(A)
& v1_finset_1(A)
& v1_fraenkel(A) ) ).
fof(fc1_closure2,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_relat_1(B)
& v1_funct_1(B) )
=> ( ~ v1_xboole_0(k2_tarski(A,B))
& v1_finset_1(k2_tarski(A,B))
& v1_fraenkel(k2_tarski(A,B)) ) ) ).
fof(fc2_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ( ~ v1_xboole_0(k1_closure2(A,B))
& v1_fraenkel(k1_closure2(A,B))
& v1_pralg_2(k1_closure2(A,B)) ) ) ).
fof(rc2_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ? [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
& ~ v1_xboole_0(C)
& v1_fraenkel(C)
& v1_pralg_2(C) ) ) ).
fof(rc3_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ? [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
& v1_xboole_0(C)
& v1_relat_1(C)
& v1_funct_1(C)
& v2_funct_1(C)
& v1_finset_1(C)
& v1_fraenkel(C) ) ) ).
fof(fc3_closure2,axiom,
! [A] :
( ( v1_xboole_0(A)
& v1_fraenkel(A) )
=> ( v1_xboole_0(k3_closure2(A))
& v1_relat_1(k3_closure2(A))
& v1_funct_1(k3_closure2(A))
& v2_funct_1(k3_closure2(A))
& v1_finset_1(k3_closure2(A))
& v1_fraenkel(k3_closure2(A)) ) ) ).
fof(fc4_closure2,axiom,
! [A,B,C] :
( ( m1_pboole(B,A)
& v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B))) )
=> ( v1_relat_1(k4_closure2(A,B,C))
& v3_relat_1(k4_closure2(A,B,C))
& v1_funct_1(k4_closure2(A,B,C))
& v1_pre_circ(k4_closure2(A,B,C),A) ) ) ).
fof(fc5_closure2,axiom,
! [A,B,C] :
( ( m1_pboole(B,A)
& ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B))) )
=> ( v1_relat_1(k4_closure2(A,B,C))
& v2_relat_1(k4_closure2(A,B,C))
& v1_funct_1(k4_closure2(A,B,C)) ) ) ).
fof(rc4_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ? [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
& ~ v1_xboole_0(C)
& v1_fraenkel(C)
& v1_pralg_2(C)
& v1_closure2(C,A,B)
& v2_closure2(C,A,B)
& v3_closure2(C,A,B)
& v4_closure2(C,A,B)
& v5_closure2(C,A,B)
& v6_closure2(C,A,B) ) ) ).
fof(cc2_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
=> ( v2_closure2(C,A,B)
=> ( v1_fraenkel(C)
& v1_closure2(C,A,B) ) ) ) ) ).
fof(cc3_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
=> ( v4_closure2(C,A,B)
=> ( v1_fraenkel(C)
& v3_closure2(C,A,B) ) ) ) ) ).
fof(cc4_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
=> ( v4_closure2(C,A,B)
=> ( v1_fraenkel(C)
& v5_closure2(C,A,B) ) ) ) ) ).
fof(cc5_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
=> ( v5_closure2(C,A,B)
=> ( ~ v1_xboole_0(C)
& v1_fraenkel(C) ) ) ) ) ).
fof(cc6_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
=> ( v2_closure2(C,A,B)
=> ( v1_fraenkel(C)
& v6_closure2(C,A,B) ) ) ) ) ).
fof(cc7_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
=> ( v6_closure2(C,A,B)
=> ( ~ v1_xboole_0(C)
& v1_fraenkel(C) ) ) ) ) ).
fof(rc5_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ? [C] :
( m1_relset_1(C,k6_closure2(A,B),k6_closure2(A,B))
& ~ v1_xboole_0(C)
& v1_relat_1(C)
& v1_funct_1(C)
& v1_funct_2(C,k6_closure2(A,B),k6_closure2(A,B))
& v1_partfun1(C,k6_closure2(A,B),k6_closure2(A,B))
& v7_closure2(C,A,B)
& v8_closure2(C,A,B)
& v9_closure2(C,A,B)
& v10_closure2(C,A,B) ) ) ).
fof(cc8_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_relset_1(C,k6_closure2(A,B),k6_closure2(A,B))
=> ( ( v1_funct_1(C)
& v1_funct_2(C,k6_closure2(A,B),k6_closure2(A,B))
& v10_closure2(C,A,B) )
=> ( ~ v1_xboole_0(C)
& v1_funct_1(C)
& v1_funct_2(C,k6_closure2(A,B),k6_closure2(A,B))
& v1_partfun1(C,k6_closure2(A,B),k6_closure2(A,B))
& v8_closure2(C,A,B) ) ) ) ) ).
fof(rc6_closure2,axiom,
! [A] :
( l1_struct_0(A)
=> ? [B] :
( l1_closure2(B,A)
& v11_closure2(B,A) ) ) ).
fof(fc6_closure2,axiom,
! [A,B] :
( ( l1_struct_0(A)
& l2_msualg_1(B,A) )
=> ( v11_closure2(k9_closure2(A,B),A)
& v12_closure2(k9_closure2(A,B),A)
& v13_closure2(k9_closure2(A,B),A)
& v14_closure2(k9_closure2(A,B),A)
& v15_closure2(k9_closure2(A,B),A)
& v16_closure2(k9_closure2(A,B),A)
& v17_closure2(k9_closure2(A,B),A) ) ) ).
fof(fc7_closure2,axiom,
! [A,B] :
( ( l1_struct_0(A)
& v5_msualg_1(B,A)
& l2_msualg_1(B,A) )
=> ( v5_msualg_1(k9_closure2(A,B),A)
& v11_closure2(k9_closure2(A,B),A)
& v12_closure2(k9_closure2(A,B),A)
& v13_closure2(k9_closure2(A,B),A)
& v14_closure2(k9_closure2(A,B),A)
& v15_closure2(k9_closure2(A,B),A)
& v16_closure2(k9_closure2(A,B),A)
& v17_closure2(k9_closure2(A,B),A) ) ) ).
fof(rc7_closure2,axiom,
! [A] :
( l1_struct_0(A)
=> ? [B] :
( l1_closure2(B,A)
& v5_msualg_1(B,A)
& v11_closure2(B,A)
& v12_closure2(B,A)
& v13_closure2(B,A)
& v14_closure2(B,A)
& v15_closure2(B,A)
& v16_closure2(B,A)
& v17_closure2(B,A) ) ) ).
fof(fc8_closure2,axiom,
! [A,B] :
( ( l1_struct_0(A)
& v12_closure2(B,A)
& l1_closure2(B,A) )
=> ( v1_fraenkel(u1_closure2(A,B))
& v1_closure2(u1_closure2(A,B),u1_struct_0(A),u4_msualg_1(A,B)) ) ) ).
fof(fc9_closure2,axiom,
! [A,B] :
( ( l1_struct_0(A)
& v13_closure2(B,A)
& l1_closure2(B,A) )
=> ( ~ v1_xboole_0(u1_closure2(A,B))
& v1_fraenkel(u1_closure2(A,B))
& v1_closure2(u1_closure2(A,B),u1_struct_0(A),u4_msualg_1(A,B))
& v2_closure2(u1_closure2(A,B),u1_struct_0(A),u4_msualg_1(A,B))
& v6_closure2(u1_closure2(A,B),u1_struct_0(A),u4_msualg_1(A,B)) ) ) ).
fof(fc10_closure2,axiom,
! [A,B] :
( ( l1_struct_0(A)
& v14_closure2(B,A)
& l1_closure2(B,A) )
=> ( v1_fraenkel(u1_closure2(A,B))
& v3_closure2(u1_closure2(A,B),u1_struct_0(A),u4_msualg_1(A,B)) ) ) ).
fof(fc11_closure2,axiom,
! [A,B] :
( ( l1_struct_0(A)
& v15_closure2(B,A)
& l1_closure2(B,A) )
=> ( ~ v1_xboole_0(u1_closure2(A,B))
& v1_fraenkel(u1_closure2(A,B))
& v3_closure2(u1_closure2(A,B),u1_struct_0(A),u4_msualg_1(A,B))
& v4_closure2(u1_closure2(A,B),u1_struct_0(A),u4_msualg_1(A,B))
& v5_closure2(u1_closure2(A,B),u1_struct_0(A),u4_msualg_1(A,B)) ) ) ).
fof(fc12_closure2,axiom,
! [A,B] :
( ( l1_struct_0(A)
& v16_closure2(B,A)
& l1_closure2(B,A) )
=> ( ~ v1_xboole_0(u1_closure2(A,B))
& v1_fraenkel(u1_closure2(A,B))
& v5_closure2(u1_closure2(A,B),u1_struct_0(A),u4_msualg_1(A,B)) ) ) ).
fof(fc13_closure2,axiom,
! [A,B] :
( ( l1_struct_0(A)
& v17_closure2(B,A)
& l1_closure2(B,A) )
=> ( ~ v1_xboole_0(u1_closure2(A,B))
& v1_fraenkel(u1_closure2(A,B))
& v6_closure2(u1_closure2(A,B),u1_struct_0(A),u4_msualg_1(A,B)) ) ) ).
fof(fc14_closure2,axiom,
! [A,B,C] :
( ( l1_struct_0(A)
& v2_relat_1(B)
& m1_pboole(B,u1_struct_0(A))
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(u1_struct_0(A),B))) )
=> ( v5_msualg_1(g1_closure2(A,B,C),A)
& v11_closure2(g1_closure2(A,B,C),A) ) ) ).
fof(fc15_closure2,axiom,
! [A,B,C] :
( ( l1_struct_0(A)
& l2_msualg_1(B,A)
& v1_closure2(C,u1_struct_0(A),u4_msualg_1(A,B))
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(u1_struct_0(A),u4_msualg_1(A,B)))) )
=> ( v11_closure2(g1_closure2(A,u4_msualg_1(A,B),C),A)
& v12_closure2(g1_closure2(A,u4_msualg_1(A,B),C),A) ) ) ).
fof(fc16_closure2,axiom,
! [A,B,C] :
( ( l1_struct_0(A)
& l2_msualg_1(B,A)
& v2_closure2(C,u1_struct_0(A),u4_msualg_1(A,B))
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(u1_struct_0(A),u4_msualg_1(A,B)))) )
=> ( v11_closure2(g1_closure2(A,u4_msualg_1(A,B),C),A)
& v12_closure2(g1_closure2(A,u4_msualg_1(A,B),C),A)
& v13_closure2(g1_closure2(A,u4_msualg_1(A,B),C),A) ) ) ).
fof(fc17_closure2,axiom,
! [A,B,C] :
( ( l1_struct_0(A)
& l2_msualg_1(B,A)
& v3_closure2(C,u1_struct_0(A),u4_msualg_1(A,B))
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(u1_struct_0(A),u4_msualg_1(A,B)))) )
=> ( v11_closure2(g1_closure2(A,u4_msualg_1(A,B),C),A)
& v14_closure2(g1_closure2(A,u4_msualg_1(A,B),C),A) ) ) ).
fof(fc18_closure2,axiom,
! [A,B,C] :
( ( l1_struct_0(A)
& l2_msualg_1(B,A)
& v4_closure2(C,u1_struct_0(A),u4_msualg_1(A,B))
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(u1_struct_0(A),u4_msualg_1(A,B)))) )
=> ( v11_closure2(g1_closure2(A,u4_msualg_1(A,B),C),A)
& v14_closure2(g1_closure2(A,u4_msualg_1(A,B),C),A)
& v15_closure2(g1_closure2(A,u4_msualg_1(A,B),C),A) ) ) ).
fof(fc19_closure2,axiom,
! [A,B,C] :
( ( l1_struct_0(A)
& l2_msualg_1(B,A)
& v5_closure2(C,u1_struct_0(A),u4_msualg_1(A,B))
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(u1_struct_0(A),u4_msualg_1(A,B)))) )
=> ( v11_closure2(g1_closure2(A,u4_msualg_1(A,B),C),A)
& v16_closure2(g1_closure2(A,u4_msualg_1(A,B),C),A) ) ) ).
fof(fc20_closure2,axiom,
! [A,B,C] :
( ( l1_struct_0(A)
& l2_msualg_1(B,A)
& v6_closure2(C,u1_struct_0(A),u4_msualg_1(A,B))
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(u1_struct_0(A),u4_msualg_1(A,B)))) )
=> ( v11_closure2(g1_closure2(A,u4_msualg_1(A,B),C),A)
& v17_closure2(g1_closure2(A,u4_msualg_1(A,B),C),A) ) ) ).
fof(cc9_closure2,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( l1_closure2(B,A)
=> ( v13_closure2(B,A)
=> v12_closure2(B,A) ) ) ) ).
fof(cc10_closure2,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( l1_closure2(B,A)
=> ( v15_closure2(B,A)
=> v14_closure2(B,A) ) ) ) ).
fof(cc11_closure2,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( l1_closure2(B,A)
=> ( v15_closure2(B,A)
=> v16_closure2(B,A) ) ) ) ).
fof(cc12_closure2,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( l1_closure2(B,A)
=> ( v13_closure2(B,A)
=> v17_closure2(B,A) ) ) ) ).
fof(t1_closure2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> ( r1_tarski(B,C)
=> r1_tarski(k8_funct_2(A,A,k6_partfun1(A),B),k8_funct_2(A,A,k6_partfun1(A),C)) ) ) ) ) ).
fof(t2_closure2,axiom,
$true ).
fof(t3_closure2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_pboole(B,A)
=> ! [C] :
( m4_pboole(C,A,B)
=> r1_tarski(k2_relat_1(C),k3_tarski(k2_relat_1(k7_mssubfam(A,B)))) ) ) ) ).
fof(d1_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( C = k1_closure2(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> m4_pboole(D,A,B) ) ) ) ).
fof(t4_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B)))
=> m1_subset_1(k4_subset_1(k1_closure2(A,B),C,D),k1_zfmisc_1(k1_closure2(A,B))) ) ) ) ).
fof(t5_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B)))
=> m1_subset_1(k5_subset_1(k1_closure2(A,B),C,D),k1_zfmisc_1(k1_closure2(A,B))) ) ) ) ).
fof(t6_closure2,axiom,
! [A,B,C] :
( m1_pboole(C,A)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,C)))
=> m1_subset_1(k4_xboole_0(D,B),k1_zfmisc_1(k1_closure2(A,C))) ) ) ).
fof(t7_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B)))
=> m1_subset_1(k7_subset_1(k1_closure2(A,B),C,D),k1_zfmisc_1(k1_closure2(A,B))) ) ) ) ).
fof(t8_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ( r2_pboole(A,B,C)
=> m1_subset_1(k1_tarski(B),k1_zfmisc_1(k1_closure2(A,C))) ) ) ) ).
fof(t9_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ! [D] :
( m1_pboole(D,A)
=> ( ( r2_pboole(A,B,C)
& r2_pboole(A,D,C) )
=> m1_subset_1(k2_tarski(B,D),k1_zfmisc_1(k1_closure2(A,C))) ) ) ) ) ).
fof(t10_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_closure2(C,A,B,k2_closure2(A,B))
=> ! [D] :
( m1_closure2(D,A,B,k2_closure2(A,B))
=> r2_hidden(k3_pboole(A,C,D),k2_closure2(A,B)) ) ) ) ).
fof(t11_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_closure2(C,A,B,k2_closure2(A,B))
=> ! [D] :
( m1_closure2(D,A,B,k2_closure2(A,B))
=> r2_hidden(k2_pboole(A,C,D),k2_closure2(A,B)) ) ) ) ).
fof(t12_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ! [D] :
( m1_closure2(D,A,C,k2_closure2(A,C))
=> r2_hidden(k4_pboole(A,D,B),k2_closure2(A,C)) ) ) ) ).
fof(t13_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_closure2(C,A,B,k2_closure2(A,B))
=> ! [D] :
( m1_closure2(D,A,B,k2_closure2(A,B))
=> r2_hidden(k5_pboole(A,C,D),k2_closure2(A,B)) ) ) ) ).
fof(d2_closure2,axiom,
$true ).
fof(t14_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B))) )
=> k1_relat_1(k3_closure2(C)) = A ) ) ).
fof(d4_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
=> ( ( C != k1_xboole_0
=> k4_closure2(A,B,C) = k3_closure2(C) )
& ( C = k1_xboole_0
=> k4_closure2(A,B,C) = k1_pboole(A) ) ) ) ) ).
fof(t16_closure2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> k1_relat_1(k3_closure2(k1_tarski(A))) = k1_relat_1(A) ) ).
fof(t17_closure2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> k1_relat_1(k3_closure2(k2_tarski(A,B))) = k3_xboole_0(k1_relat_1(A),k1_relat_1(B)) ) ) ).
fof(t18_closure2,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r2_hidden(A,k1_relat_1(B))
=> k1_funct_1(k3_closure2(k1_tarski(B)),A) = k1_tarski(k1_funct_1(B,A)) ) ) ).
fof(t19_closure2,axiom,
! [A,B,C] :
( m1_pboole(C,B)
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(k1_closure2(B,C)))
=> ( ( r2_hidden(A,B)
& E = k1_tarski(D) )
=> k1_funct_1(k4_closure2(B,C,E),A) = k1_tarski(k1_funct_1(D,A)) ) ) ) ) ).
fof(t20_closure2,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( r2_hidden(A,k1_relat_1(k3_closure2(k2_tarski(B,C))))
=> k1_funct_1(k3_closure2(k2_tarski(B,C)),A) = k2_tarski(k1_funct_1(B,A),k1_funct_1(C,A)) ) ) ) ).
fof(t21_closure2,axiom,
! [A,B,C] :
( m1_pboole(C,B)
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ! [E] :
( ( v1_relat_1(E)
& v1_funct_1(E) )
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(k1_closure2(B,C)))
=> ( ( r2_hidden(A,B)
& F = k2_tarski(D,E) )
=> k1_funct_1(k4_closure2(B,C,F),A) = k2_tarski(k1_funct_1(D,A),k1_funct_1(E,A)) ) ) ) ) ) ).
fof(t22_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B)))
=> ( r2_hidden(C,D)
=> r1_pboole(A,C,k5_closure2(A,B,D)) ) ) ) ) ).
fof(t23_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ! [D] :
( m1_pboole(D,A)
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(k1_closure2(A,B)))
=> ( E = k2_tarski(C,D)
=> r6_pboole(A,k2_mboolean(A,k5_closure2(A,B,E)),k2_pboole(A,C,D)) ) ) ) ) ) ).
fof(t24_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
=> ! [D] :
( m1_closure2(D,A,B,k2_closure2(A,B))
=> ! [E] :
( m1_closure2(E,A,B,k2_closure2(A,B))
=> ( C = k2_tarski(D,E)
=> r6_pboole(A,k6_mssubfam(A,B,k5_closure2(A,B,C)),k3_pboole(A,D,E)) ) ) ) ) ) ).
fof(t25_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
=> ! [D] :
( m4_pboole(D,A,B)
=> ( ! [E] :
( m1_pboole(E,A)
=> ( r2_hidden(E,C)
=> r2_pboole(A,D,E) ) )
=> r2_pboole(A,D,k6_mssubfam(A,B,k5_closure2(A,B,C))) ) ) ) ) ).
fof(t26_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> r6_pboole(A,k5_closure2(A,B,k2_closure2(A,B)),k7_mssubfam(A,B)) ) ).
fof(d5_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
=> ( v1_closure2(C,A,B)
<=> ! [D] :
( m1_pboole(D,A)
=> ! [E] :
( m1_pboole(E,A)
=> ( ( r2_hidden(D,C)
& r2_hidden(E,C) )
=> r2_hidden(k2_pboole(A,D,E),C) ) ) ) ) ) ) ).
fof(d6_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
=> ( v2_closure2(C,A,B)
<=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B)))
=> ( r1_tarski(D,C)
=> r2_hidden(k2_mboolean(A,k5_closure2(A,B,D)),C) ) ) ) ) ) ).
fof(d7_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
=> ( v3_closure2(C,A,B)
<=> ! [D] :
( m1_pboole(D,A)
=> ! [E] :
( m1_pboole(E,A)
=> ( ( r2_hidden(D,C)
& r2_hidden(E,C) )
=> r2_hidden(k3_pboole(A,D,E),C) ) ) ) ) ) ) ).
fof(d8_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
=> ( v4_closure2(C,A,B)
<=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B)))
=> ( r1_tarski(D,C)
=> r2_hidden(k6_mssubfam(A,B,k5_closure2(A,B,D)),C) ) ) ) ) ) ).
fof(d9_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
=> ( v5_closure2(C,A,B)
<=> r2_hidden(B,C) ) ) ) ).
fof(d10_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
=> ( v6_closure2(C,A,B)
<=> r2_hidden(k1_pboole(A),C) ) ) ) ).
fof(d11_closure2,axiom,
$true ).
fof(d12_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k6_closure2(A,B),k6_closure2(A,B))
& m2_relset_1(C,k6_closure2(A,B),k6_closure2(A,B)) )
=> ( v7_closure2(C,A,B)
<=> ! [D] :
( m1_closure2(D,A,B,k6_closure2(A,B))
=> r2_pboole(A,D,k7_closure2(A,B,C,D)) ) ) ) ) ).
fof(d13_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k6_closure2(A,B),k6_closure2(A,B))
& m2_relset_1(C,k6_closure2(A,B),k6_closure2(A,B)) )
=> ( v8_closure2(C,A,B)
<=> ! [D] :
( m1_closure2(D,A,B,k6_closure2(A,B))
=> ! [E] :
( m1_closure2(E,A,B,k6_closure2(A,B))
=> ( r2_pboole(A,D,E)
=> r2_pboole(A,k7_closure2(A,B,C,D),k7_closure2(A,B,C,E)) ) ) ) ) ) ) ).
fof(d14_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k6_closure2(A,B),k6_closure2(A,B))
& m2_relset_1(C,k6_closure2(A,B),k6_closure2(A,B)) )
=> ( v9_closure2(C,A,B)
<=> ! [D] :
( m1_closure2(D,A,B,k6_closure2(A,B))
=> r6_pboole(A,k7_closure2(A,B,C,D),k7_closure2(A,B,C,k7_closure2(A,B,C,D))) ) ) ) ) ).
fof(d15_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k6_closure2(A,B),k6_closure2(A,B))
& m2_relset_1(C,k6_closure2(A,B),k6_closure2(A,B)) )
=> ( v10_closure2(C,A,B)
<=> ! [D] :
( m1_closure2(D,A,B,k6_closure2(A,B))
=> ! [E] :
( m1_closure2(E,A,B,k6_closure2(A,B))
=> k1_funct_1(C,k2_pboole(A,D,E)) = k2_pboole(A,k7_closure2(A,B,C,D),k7_closure2(A,B,C,E)) ) ) ) ) ) ).
fof(t27_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ( v1_funct_1(k6_partfun1(k6_closure2(A,B)))
& v1_funct_2(k6_partfun1(k6_closure2(A,B)),k6_closure2(A,B),k6_closure2(A,B))
& v7_closure2(k6_partfun1(k6_closure2(A,B)),A,B)
& m2_relset_1(k6_partfun1(k6_closure2(A,B)),k6_closure2(A,B),k6_closure2(A,B)) ) ) ).
fof(t28_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ( v1_funct_1(k6_partfun1(k6_closure2(A,B)))
& v1_funct_2(k6_partfun1(k6_closure2(A,B)),k6_closure2(A,B),k6_closure2(A,B))
& v8_closure2(k6_partfun1(k6_closure2(A,B)),A,B)
& m2_relset_1(k6_partfun1(k6_closure2(A,B)),k6_closure2(A,B),k6_closure2(A,B)) ) ) ).
fof(t29_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ( v1_funct_1(k6_partfun1(k6_closure2(A,B)))
& v1_funct_2(k6_partfun1(k6_closure2(A,B)),k6_closure2(A,B),k6_closure2(A,B))
& v9_closure2(k6_partfun1(k6_closure2(A,B)),A,B)
& m2_relset_1(k6_partfun1(k6_closure2(A,B)),k6_closure2(A,B),k6_closure2(A,B)) ) ) ).
fof(t30_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ( v1_funct_1(k6_partfun1(k6_closure2(A,B)))
& v1_funct_2(k6_partfun1(k6_closure2(A,B)),k6_closure2(A,B),k6_closure2(A,B))
& v10_closure2(k6_partfun1(k6_closure2(A,B)),A,B)
& m2_relset_1(k6_partfun1(k6_closure2(A,B)),k6_closure2(A,B),k6_closure2(A,B)) ) ) ).
fof(t31_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_closure2(C,A,B,k2_closure2(A,B))
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k6_closure2(A,B),k6_closure2(A,B))
& m2_relset_1(D,k6_closure2(A,B),k6_closure2(A,B)) )
=> ( ( r6_pboole(A,C,B)
& v7_closure2(D,A,B) )
=> r6_pboole(A,C,k7_closure2(A,B,D,C)) ) ) ) ) ).
fof(t32_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k6_closure2(A,B),k6_closure2(A,B))
& m2_relset_1(C,k6_closure2(A,B),k6_closure2(A,B)) )
=> ( ( v7_closure2(C,A,B)
& ! [D] :
( m1_closure2(D,A,B,k6_closure2(A,B))
=> r2_pboole(A,k7_closure2(A,B,C,D),D) ) )
=> v9_closure2(C,A,B) ) ) ) ).
fof(t33_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_closure2(C,A,B,k2_closure2(A,B))
=> ! [D] :
( m1_closure2(D,A,B,k2_closure2(A,B))
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k6_closure2(A,B),k6_closure2(A,B))
& m2_relset_1(E,k6_closure2(A,B),k6_closure2(A,B)) )
=> ! [F] :
( m1_closure2(F,A,B,k6_closure2(A,B))
=> ( ( r6_pboole(A,F,k3_pboole(A,C,D))
& v8_closure2(E,A,B) )
=> r2_pboole(A,k7_closure2(A,B,E,F),k3_pboole(A,k7_closure2(A,B,E,C),k7_closure2(A,B,E,D))) ) ) ) ) ) ) ).
fof(t34_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_closure2(C,A,B,k2_closure2(A,B))
=> ! [D] :
( m1_closure2(D,A,B,k2_closure2(A,B))
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k6_closure2(A,B),k6_closure2(A,B))
& m2_relset_1(E,k6_closure2(A,B),k6_closure2(A,B)) )
=> ! [F] :
( m1_closure2(F,A,B,k6_closure2(A,B))
=> ( ( r6_pboole(A,F,k4_pboole(A,C,D))
& v10_closure2(E,A,B) )
=> r2_pboole(A,k4_pboole(A,k7_closure2(A,B,E,C),k7_closure2(A,B,E,D)),k7_closure2(A,B,E,F)) ) ) ) ) ) ) ).
fof(t35_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k6_closure2(A,B),k6_closure2(A,B))
& m2_relset_1(C,k6_closure2(A,B),k6_closure2(A,B)) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k6_closure2(A,B),k6_closure2(A,B))
& m2_relset_1(D,k6_closure2(A,B),k6_closure2(A,B)) )
=> ( ( v7_closure2(C,A,B)
& v7_closure2(D,A,B) )
=> v7_closure2(k8_closure2(A,B,D,C),A,B) ) ) ) ) ).
fof(t36_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k6_closure2(A,B),k6_closure2(A,B))
& m2_relset_1(C,k6_closure2(A,B),k6_closure2(A,B)) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k6_closure2(A,B),k6_closure2(A,B))
& m2_relset_1(D,k6_closure2(A,B),k6_closure2(A,B)) )
=> ( ( v8_closure2(C,A,B)
& v8_closure2(D,A,B) )
=> v8_closure2(k8_closure2(A,B,D,C),A,B) ) ) ) ) ).
fof(t37_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k6_closure2(A,B),k6_closure2(A,B))
& m2_relset_1(C,k6_closure2(A,B),k6_closure2(A,B)) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k6_closure2(A,B),k6_closure2(A,B))
& m2_relset_1(D,k6_closure2(A,B),k6_closure2(A,B)) )
=> ( ( v9_closure2(C,A,B)
& v9_closure2(D,A,B)
& k8_closure2(A,B,D,C) = k8_closure2(A,B,C,D) )
=> v9_closure2(k8_closure2(A,B,D,C),A,B) ) ) ) ) ).
fof(t38_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k6_closure2(A,B),k6_closure2(A,B))
& m2_relset_1(C,k6_closure2(A,B),k6_closure2(A,B)) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k6_closure2(A,B),k6_closure2(A,B))
& m2_relset_1(D,k6_closure2(A,B),k6_closure2(A,B)) )
=> ( ( v10_closure2(C,A,B)
& v10_closure2(D,A,B) )
=> v10_closure2(k8_closure2(A,B,D,C),A,B) ) ) ) ) ).
fof(d16_closure2,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( l1_closure2(B,A)
=> ( v12_closure2(B,A)
<=> v1_closure2(u1_closure2(A,B),u1_struct_0(A),u4_msualg_1(A,B)) ) ) ) ).
fof(d17_closure2,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( l1_closure2(B,A)
=> ( v13_closure2(B,A)
<=> v2_closure2(u1_closure2(A,B),u1_struct_0(A),u4_msualg_1(A,B)) ) ) ) ).
fof(d18_closure2,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( l1_closure2(B,A)
=> ( v14_closure2(B,A)
<=> v3_closure2(u1_closure2(A,B),u1_struct_0(A),u4_msualg_1(A,B)) ) ) ) ).
fof(d19_closure2,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( l1_closure2(B,A)
=> ( v15_closure2(B,A)
<=> v4_closure2(u1_closure2(A,B),u1_struct_0(A),u4_msualg_1(A,B)) ) ) ) ).
fof(d20_closure2,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( l1_closure2(B,A)
=> ( v16_closure2(B,A)
<=> v5_closure2(u1_closure2(A,B),u1_struct_0(A),u4_msualg_1(A,B)) ) ) ) ).
fof(d21_closure2,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( l1_closure2(B,A)
=> ( v17_closure2(B,A)
<=> v6_closure2(u1_closure2(A,B),u1_struct_0(A),u4_msualg_1(A,B)) ) ) ) ).
fof(d22_closure2,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( l2_msualg_1(B,A)
=> k9_closure2(A,B) = g1_closure2(A,u4_msualg_1(A,B),k6_closure2(u1_struct_0(A),u4_msualg_1(A,B))) ) ) ).
fof(t40_closure2,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( ( v15_closure2(B,A)
& l1_closure2(B,A) )
=> ! [C] :
( m1_closure2(C,u1_struct_0(A),u4_msualg_1(A,B),k6_closure2(u1_struct_0(A),u4_msualg_1(A,B)))
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k6_closure2(u1_struct_0(A),u4_msualg_1(A,B)),k6_closure2(u1_struct_0(A),u4_msualg_1(A,B)))
& m2_relset_1(D,k6_closure2(u1_struct_0(A),u4_msualg_1(A,B)),k6_closure2(u1_struct_0(A),u4_msualg_1(A,B))) )
=> ( ( r2_hidden(C,u1_closure2(A,B))
& ! [E] :
( m1_closure2(E,u1_struct_0(A),u4_msualg_1(A,B),k6_closure2(u1_struct_0(A),u4_msualg_1(A,B)))
=> r6_pboole(u1_struct_0(A),k7_closure2(u1_struct_0(A),u4_msualg_1(A,B),D,E),k11_closure2(A,B,E)) ) )
=> r6_pboole(u1_struct_0(A),k7_closure2(u1_struct_0(A),u4_msualg_1(A,B),D,C),C) ) ) ) ) ) ).
fof(t41_closure2,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( ( v15_closure2(B,A)
& l1_closure2(B,A) )
=> ! [C] :
( m1_closure2(C,u1_struct_0(A),u4_msualg_1(A,B),k6_closure2(u1_struct_0(A),u4_msualg_1(A,B)))
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k6_closure2(u1_struct_0(A),u4_msualg_1(A,B)),k6_closure2(u1_struct_0(A),u4_msualg_1(A,B)))
& m2_relset_1(D,k6_closure2(u1_struct_0(A),u4_msualg_1(A,B)),k6_closure2(u1_struct_0(A),u4_msualg_1(A,B))) )
=> ( ( r6_pboole(u1_struct_0(A),k7_closure2(u1_struct_0(A),u4_msualg_1(A,B),D,C),C)
& ! [E] :
( m1_closure2(E,u1_struct_0(A),u4_msualg_1(A,B),k6_closure2(u1_struct_0(A),u4_msualg_1(A,B)))
=> r6_pboole(u1_struct_0(A),k7_closure2(u1_struct_0(A),u4_msualg_1(A,B),D,E),k11_closure2(A,B,E)) ) )
=> r2_hidden(C,u1_closure2(A,B)) ) ) ) ) ) ).
fof(t42_closure2,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( ( v15_closure2(B,A)
& l1_closure2(B,A) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k6_closure2(u1_struct_0(A),u4_msualg_1(A,B)),k6_closure2(u1_struct_0(A),u4_msualg_1(A,B)))
& m2_relset_1(C,k6_closure2(u1_struct_0(A),u4_msualg_1(A,B)),k6_closure2(u1_struct_0(A),u4_msualg_1(A,B))) )
=> ( ! [D] :
( m1_closure2(D,u1_struct_0(A),u4_msualg_1(A,B),k6_closure2(u1_struct_0(A),u4_msualg_1(A,B)))
=> r6_pboole(u1_struct_0(A),k7_closure2(u1_struct_0(A),u4_msualg_1(A,B),C,D),k11_closure2(A,B,D)) )
=> ( v7_closure2(C,u1_struct_0(A),u4_msualg_1(A,B))
& v8_closure2(C,u1_struct_0(A),u4_msualg_1(A,B))
& v9_closure2(C,u1_struct_0(A),u4_msualg_1(A,B)) ) ) ) ) ) ).
fof(d25_closure2,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( ( v15_closure2(B,A)
& l1_closure2(B,A) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k6_closure2(u1_struct_0(A),u4_msualg_1(A,B)),k6_closure2(u1_struct_0(A),u4_msualg_1(A,B)))
& v7_closure2(C,u1_struct_0(A),u4_msualg_1(A,B))
& v8_closure2(C,u1_struct_0(A),u4_msualg_1(A,B))
& v9_closure2(C,u1_struct_0(A),u4_msualg_1(A,B))
& m2_relset_1(C,k6_closure2(u1_struct_0(A),u4_msualg_1(A,B)),k6_closure2(u1_struct_0(A),u4_msualg_1(A,B))) )
=> ( C = k12_closure2(A,B)
<=> ! [D] :
( m1_closure2(D,u1_struct_0(A),u4_msualg_1(A,B),k6_closure2(u1_struct_0(A),u4_msualg_1(A,B)))
=> r6_pboole(u1_struct_0(A),k7_closure2(u1_struct_0(A),u4_msualg_1(A,B),C,D),k11_closure2(A,B,D)) ) ) ) ) ) ).
fof(t43_closure2,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( m1_pboole(B,u1_struct_0(A))
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k6_closure2(u1_struct_0(A),B),k6_closure2(u1_struct_0(A),B))
& v7_closure2(C,u1_struct_0(A),B)
& v8_closure2(C,u1_struct_0(A),B)
& v9_closure2(C,u1_struct_0(A),B)
& m2_relset_1(C,k6_closure2(u1_struct_0(A),B),k6_closure2(u1_struct_0(A),B)) )
=> k12_closure2(A,k10_closure2(A,B,C)) = C ) ) ) ).
fof(t44_closure2,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( ( v15_closure2(B,A)
& l1_closure2(B,A) )
=> k10_closure2(A,u4_msualg_1(A,B),k12_closure2(A,B)) = g1_closure2(A,u4_msualg_1(A,B),u1_closure2(A,B)) ) ) ).
fof(dt_m1_closure2,axiom,
! [A,B,C] :
( ( m1_pboole(B,A)
& ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B))) )
=> ! [D] :
( m1_closure2(D,A,B,C)
=> m4_pboole(D,A,B) ) ) ).
fof(existence_m1_closure2,axiom,
! [A,B,C] :
( ( m1_pboole(B,A)
& ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B))) )
=> ? [D] : m1_closure2(D,A,B,C) ) ).
fof(redefinition_m1_closure2,axiom,
! [A,B,C] :
( ( m1_pboole(B,A)
& ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B))) )
=> ! [D] :
( m1_closure2(D,A,B,C)
<=> m1_subset_1(D,C) ) ) ).
fof(dt_l1_closure2,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( l1_closure2(B,A)
=> l2_msualg_1(B,A) ) ) ).
fof(existence_l1_closure2,axiom,
! [A] :
( l1_struct_0(A)
=> ? [B] : l1_closure2(B,A) ) ).
fof(abstractness_v11_closure2,axiom,
! [A,B] :
( ( l1_struct_0(A)
& l1_closure2(B,A) )
=> ( v11_closure2(B,A)
=> B = g1_closure2(A,u4_msualg_1(A,B),u1_closure2(A,B)) ) ) ).
fof(dt_k1_closure2,axiom,
$true ).
fof(dt_k2_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> m1_subset_1(k2_closure2(A,B),k1_zfmisc_1(k1_closure2(A,B))) ) ).
fof(redefinition_k2_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> k2_closure2(A,B) = k1_closure2(A,B) ) ).
fof(dt_k3_closure2,axiom,
! [A] :
( v1_fraenkel(A)
=> ( v1_relat_1(k3_closure2(A))
& v1_funct_1(k3_closure2(A)) ) ) ).
fof(dt_k4_closure2,axiom,
! [A,B,C] :
( ( m1_pboole(B,A)
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B))) )
=> m1_pboole(k4_closure2(A,B,C),A) ) ).
fof(dt_k5_closure2,axiom,
! [A,B,C] :
( ( m1_pboole(B,A)
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B))) )
=> m4_pboole(k5_closure2(A,B,C),A,k1_mboolean(A,B)) ) ).
fof(redefinition_k5_closure2,axiom,
! [A,B,C] :
( ( m1_pboole(B,A)
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B))) )
=> k5_closure2(A,B,C) = k4_closure2(A,B,C) ) ).
fof(dt_k6_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ( v1_closure2(k6_closure2(A,B),A,B)
& v2_closure2(k6_closure2(A,B),A,B)
& v3_closure2(k6_closure2(A,B),A,B)
& v4_closure2(k6_closure2(A,B),A,B)
& v5_closure2(k6_closure2(A,B),A,B)
& v6_closure2(k6_closure2(A,B),A,B)
& m1_subset_1(k6_closure2(A,B),k1_zfmisc_1(k1_closure2(A,B))) ) ) ).
fof(redefinition_k6_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> k6_closure2(A,B) = k1_closure2(A,B) ) ).
fof(dt_k7_closure2,axiom,
! [A,B,C,D] :
( ( m1_pboole(B,A)
& v1_funct_1(C)
& v1_funct_2(C,k6_closure2(A,B),k6_closure2(A,B))
& m1_relset_1(C,k6_closure2(A,B),k6_closure2(A,B))
& m1_subset_1(D,k6_closure2(A,B)) )
=> m1_closure2(k7_closure2(A,B,C,D),A,B,k6_closure2(A,B)) ) ).
fof(redefinition_k7_closure2,axiom,
! [A,B,C,D] :
( ( m1_pboole(B,A)
& v1_funct_1(C)
& v1_funct_2(C,k6_closure2(A,B),k6_closure2(A,B))
& m1_relset_1(C,k6_closure2(A,B),k6_closure2(A,B))
& m1_subset_1(D,k6_closure2(A,B)) )
=> k7_closure2(A,B,C,D) = k1_funct_1(C,D) ) ).
fof(dt_k8_closure2,axiom,
! [A,B,C,D] :
( ( m1_pboole(B,A)
& v1_funct_1(C)
& v1_funct_2(C,k6_closure2(A,B),k6_closure2(A,B))
& m1_relset_1(C,k6_closure2(A,B),k6_closure2(A,B))
& v1_funct_1(D)
& v1_funct_2(D,k6_closure2(A,B),k6_closure2(A,B))
& m1_relset_1(D,k6_closure2(A,B),k6_closure2(A,B)) )
=> ( v1_funct_1(k8_closure2(A,B,C,D))
& v1_funct_2(k8_closure2(A,B,C,D),k6_closure2(A,B),k6_closure2(A,B))
& m2_relset_1(k8_closure2(A,B,C,D),k6_closure2(A,B),k6_closure2(A,B)) ) ) ).
fof(redefinition_k8_closure2,axiom,
! [A,B,C,D] :
( ( m1_pboole(B,A)
& v1_funct_1(C)
& v1_funct_2(C,k6_closure2(A,B),k6_closure2(A,B))
& m1_relset_1(C,k6_closure2(A,B),k6_closure2(A,B))
& v1_funct_1(D)
& v1_funct_2(D,k6_closure2(A,B),k6_closure2(A,B))
& m1_relset_1(D,k6_closure2(A,B),k6_closure2(A,B)) )
=> k8_closure2(A,B,C,D) = k5_relat_1(C,D) ) ).
fof(dt_k9_closure2,axiom,
! [A,B] :
( ( l1_struct_0(A)
& l2_msualg_1(B,A) )
=> l1_closure2(k9_closure2(A,B),A) ) ).
fof(dt_k10_closure2,axiom,
! [A,B,C] :
( ( l1_struct_0(A)
& m1_pboole(B,u1_struct_0(A))
& v1_funct_1(C)
& v1_funct_2(C,k6_closure2(u1_struct_0(A),B),k6_closure2(u1_struct_0(A),B))
& v7_closure2(C,u1_struct_0(A),B)
& v8_closure2(C,u1_struct_0(A),B)
& v9_closure2(C,u1_struct_0(A),B)
& m1_relset_1(C,k6_closure2(u1_struct_0(A),B),k6_closure2(u1_struct_0(A),B)) )
=> ( v11_closure2(k10_closure2(A,B,C),A)
& v15_closure2(k10_closure2(A,B,C),A)
& l1_closure2(k10_closure2(A,B,C),A) ) ) ).
fof(dt_k11_closure2,axiom,
! [A,B,C] :
( ( l1_struct_0(A)
& v15_closure2(B,A)
& l1_closure2(B,A)
& m4_pboole(C,u1_struct_0(A),u4_msualg_1(A,B)) )
=> m1_closure2(k11_closure2(A,B,C),u1_struct_0(A),u4_msualg_1(A,B),k6_closure2(u1_struct_0(A),u4_msualg_1(A,B))) ) ).
fof(dt_k12_closure2,axiom,
! [A,B] :
( ( l1_struct_0(A)
& v15_closure2(B,A)
& l1_closure2(B,A) )
=> ( v1_funct_1(k12_closure2(A,B))
& v1_funct_2(k12_closure2(A,B),k6_closure2(u1_struct_0(A),u4_msualg_1(A,B)),k6_closure2(u1_struct_0(A),u4_msualg_1(A,B)))
& v7_closure2(k12_closure2(A,B),u1_struct_0(A),u4_msualg_1(A,B))
& v8_closure2(k12_closure2(A,B),u1_struct_0(A),u4_msualg_1(A,B))
& v9_closure2(k12_closure2(A,B),u1_struct_0(A),u4_msualg_1(A,B))
& m2_relset_1(k12_closure2(A,B),k6_closure2(u1_struct_0(A),u4_msualg_1(A,B)),k6_closure2(u1_struct_0(A),u4_msualg_1(A,B))) ) ) ).
fof(dt_u1_closure2,axiom,
! [A,B] :
( ( l1_struct_0(A)
& l1_closure2(B,A) )
=> m1_subset_1(u1_closure2(A,B),k1_zfmisc_1(k1_closure2(u1_struct_0(A),u4_msualg_1(A,B)))) ) ).
fof(dt_g1_closure2,axiom,
! [A,B,C] :
( ( l1_struct_0(A)
& m1_pboole(B,u1_struct_0(A))
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(u1_struct_0(A),B))) )
=> ( v11_closure2(g1_closure2(A,B,C),A)
& l1_closure2(g1_closure2(A,B,C),A) ) ) ).
fof(free_g1_closure2,axiom,
! [A,B,C] :
( ( l1_struct_0(A)
& m1_pboole(B,u1_struct_0(A))
& m1_subset_1(C,k1_zfmisc_1(k1_closure2(u1_struct_0(A),B))) )
=> ! [D,E,F] :
( g1_closure2(A,B,C) = g1_closure2(D,E,F)
=> ( A = D
& B = E
& C = F ) ) ) ).
fof(d3_closure2,axiom,
! [A] :
( v1_fraenkel(A)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( ( A != k1_xboole_0
=> ( B = k3_closure2(A)
<=> ? [C] :
( ~ v1_xboole_0(C)
& v1_fraenkel(C)
& C = A
& k1_relat_1(B) = k1_setfam_1(a_1_0_closure2(C))
& ! [D] :
( r2_hidden(D,k1_relat_1(B))
=> k1_funct_1(B,D) = a_2_0_closure2(C,D) ) ) ) )
& ( A = k1_xboole_0
=> ( B = k3_closure2(A)
<=> B = k1_xboole_0 ) ) ) ) ) ).
fof(t15_closure2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
=> ( ~ v1_xboole_0(C)
=> ! [D] :
( r2_hidden(D,A)
=> k1_funct_1(k4_closure2(A,B,C),D) = a_4_0_closure2(A,B,C,D) ) ) ) ) ).
fof(t39_closure2,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( m1_pboole(B,u1_struct_0(A))
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k6_closure2(u1_struct_0(A),B),k6_closure2(u1_struct_0(A),B))
& v7_closure2(C,u1_struct_0(A),B)
& v8_closure2(C,u1_struct_0(A),B)
& m2_relset_1(C,k6_closure2(u1_struct_0(A),B),k6_closure2(u1_struct_0(A),B)) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_closure2(u1_struct_0(A),B)))
=> ( D = a_3_0_closure2(A,B,C)
=> ( v15_closure2(g1_closure2(A,B,D),A)
& l1_closure2(g1_closure2(A,B,D),A) ) ) ) ) ) ) ).
fof(d23_closure2,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( m1_pboole(B,u1_struct_0(A))
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k6_closure2(u1_struct_0(A),B),k6_closure2(u1_struct_0(A),B))
& v7_closure2(C,u1_struct_0(A),B)
& v8_closure2(C,u1_struct_0(A),B)
& v9_closure2(C,u1_struct_0(A),B)
& m2_relset_1(C,k6_closure2(u1_struct_0(A),B),k6_closure2(u1_struct_0(A),B)) )
=> ! [D] :
( ( v11_closure2(D,A)
& v15_closure2(D,A)
& l1_closure2(D,A) )
=> ( D = k10_closure2(A,B,C)
<=> ( r6_pboole(u1_struct_0(A),u4_msualg_1(A,D),B)
& u1_closure2(A,D) = a_3_1_closure2(A,B,C) ) ) ) ) ) ) ).
fof(d24_closure2,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( ( v15_closure2(B,A)
& l1_closure2(B,A) )
=> ! [C] :
( m4_pboole(C,u1_struct_0(A),u4_msualg_1(A,B))
=> ! [D] :
( m1_closure2(D,u1_struct_0(A),u4_msualg_1(A,B),k6_closure2(u1_struct_0(A),u4_msualg_1(A,B)))
=> ( D = k11_closure2(A,B,C)
<=> ? [E] :
( m1_subset_1(E,k1_zfmisc_1(k1_closure2(u1_struct_0(A),u4_msualg_1(A,B))))
& r6_pboole(u1_struct_0(A),D,k6_mssubfam(u1_struct_0(A),u4_msualg_1(A,B),k5_closure2(u1_struct_0(A),u4_msualg_1(A,B),E)))
& E = a_3_2_closure2(A,B,C) ) ) ) ) ) ) ).
fof(fraenkel_a_1_0_closure2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v1_fraenkel(B) )
=> ( r2_hidden(A,a_1_0_closure2(B))
<=> ? [C] :
( m1_subset_1(C,B)
& A = k1_relat_1(C) ) ) ) ).
fof(fraenkel_a_2_0_closure2,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& v1_fraenkel(B) )
=> ( r2_hidden(A,a_2_0_closure2(B,C))
<=> ? [D] :
( m1_subset_1(D,B)
& A = k1_funct_1(D,C) ) ) ) ).
fof(fraenkel_a_4_0_closure2,axiom,
! [A,B,C,D,E] :
( ( m1_pboole(C,B)
& m1_subset_1(D,k1_zfmisc_1(k1_closure2(B,C))) )
=> ( r2_hidden(A,a_4_0_closure2(B,C,D,E))
<=> ? [F] :
( m1_closure2(F,B,C,k2_closure2(B,C))
& A = k1_funct_1(F,E)
& r2_hidden(F,D) ) ) ) ).
fof(fraenkel_a_3_0_closure2,axiom,
! [A,B,C,D] :
( ( l1_struct_0(B)
& m1_pboole(C,u1_struct_0(B))
& v1_funct_1(D)
& v1_funct_2(D,k6_closure2(u1_struct_0(B),C),k6_closure2(u1_struct_0(B),C))
& v7_closure2(D,u1_struct_0(B),C)
& v8_closure2(D,u1_struct_0(B),C)
& m2_relset_1(D,k6_closure2(u1_struct_0(B),C),k6_closure2(u1_struct_0(B),C)) )
=> ( r2_hidden(A,a_3_0_closure2(B,C,D))
<=> ? [E] :
( m1_closure2(E,u1_struct_0(B),C,k6_closure2(u1_struct_0(B),C))
& A = E
& r6_pboole(u1_struct_0(B),k7_closure2(u1_struct_0(B),C,D,E),E) ) ) ) ).
fof(fraenkel_a_3_1_closure2,axiom,
! [A,B,C,D] :
( ( l1_struct_0(B)
& m1_pboole(C,u1_struct_0(B))
& v1_funct_1(D)
& v1_funct_2(D,k6_closure2(u1_struct_0(B),C),k6_closure2(u1_struct_0(B),C))
& v7_closure2(D,u1_struct_0(B),C)
& v8_closure2(D,u1_struct_0(B),C)
& v9_closure2(D,u1_struct_0(B),C)
& m2_relset_1(D,k6_closure2(u1_struct_0(B),C),k6_closure2(u1_struct_0(B),C)) )
=> ( r2_hidden(A,a_3_1_closure2(B,C,D))
<=> ? [E] :
( m1_closure2(E,u1_struct_0(B),C,k6_closure2(u1_struct_0(B),C))
& A = E
& r6_pboole(u1_struct_0(B),k7_closure2(u1_struct_0(B),C,D,E),E) ) ) ) ).
fof(fraenkel_a_3_2_closure2,axiom,
! [A,B,C,D] :
( ( l1_struct_0(B)
& v15_closure2(C,B)
& l1_closure2(C,B)
& m4_pboole(D,u1_struct_0(B),u4_msualg_1(B,C)) )
=> ( r2_hidden(A,a_3_2_closure2(B,C,D))
<=> ? [E] :
( m1_closure2(E,u1_struct_0(B),u4_msualg_1(B,C),k6_closure2(u1_struct_0(B),u4_msualg_1(B,C)))
& A = E
& r2_pboole(u1_struct_0(B),D,E)
& r2_hidden(E,u1_closure2(B,C)) ) ) ) ).
%------------------------------------------------------------------------------