SET007 Axioms: SET007+295.ax
%------------------------------------------------------------------------------
% File : SET007+295 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Filters - Part II.
% Version : [Urb08] axioms.
% English : Quotient Lattices Modulo Filters and Direct Product of Two
% Lattices
% 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 : filter_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 103 ( 0 unt; 0 def)
% Number of atoms : 994 ( 46 equ)
% Maximal formula atoms : 21 ( 9 avg)
% Number of connectives : 1051 ( 160 ~; 0 |; 510 &)
% ( 29 <=>; 352 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 10 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 53 ( 52 usr; 0 prp; 1-3 aty)
% Number of functors : 57 ( 57 usr; 12 con; 0-6 aty)
% Number of variables : 367 ( 363 !; 4 ?)
% 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_filter_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m1_relset_1(B,A,A) )
=> ~ v1_xboole_0(k7_eqrel_1(A,B)) ) ).
fof(fc2_filter_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l3_lattices(A)
& ~ v3_struct_0(B)
& l3_lattices(B) )
=> ( ~ v3_struct_0(k8_filter_1(A,B))
& v3_lattices(k8_filter_1(A,B)) ) ) ).
fof(fc3_filter_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A)
& ~ v3_struct_0(B)
& v10_lattices(B)
& l3_lattices(B) )
=> ( ~ v3_struct_0(k8_filter_1(A,B))
& v3_lattices(k8_filter_1(A,B))
& v4_lattices(k8_filter_1(A,B))
& v5_lattices(k8_filter_1(A,B))
& v6_lattices(k8_filter_1(A,B))
& v7_lattices(k8_filter_1(A,B))
& v8_lattices(k8_filter_1(A,B))
& v9_lattices(k8_filter_1(A,B))
& v10_lattices(k8_filter_1(A,B)) ) ) ).
fof(t1_filter_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_filter_0(B,A)
=> ! [C] :
( m1_filter_0(C,A)
=> m1_filter_0(k5_subset_1(u1_struct_0(A),B,C),A) ) ) ) ).
fof(t2_filter_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( k2_filter_0(A,B) = k2_filter_0(A,C)
=> B = C ) ) ) ) ).
fof(d1_filter_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( v1_relat_1(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,A)
& m2_relset_1(C,A,A) )
=> ( m1_filter_1(C,A,B)
<=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,A)
=> ( r2_hidden(k1_domain_1(A,A,D,E),B)
=> r2_hidden(k1_domain_1(A,A,k8_funct_2(A,A,C,D),k8_funct_2(A,A,C,E)),B) ) ) ) ) ) ) ) ).
fof(d2_filter_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( v1_relat_1(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ( m2_filter_1(C,A,B)
<=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,A)
=> ! [F] :
( m1_subset_1(F,A)
=> ! [G] :
( m1_subset_1(G,A)
=> ( ( r2_hidden(k1_domain_1(A,A,D,E),B)
& r2_hidden(k1_domain_1(A,A,F,G),B) )
=> r2_hidden(k1_domain_1(A,A,k2_binop_1(A,A,A,C,D,F),k2_binop_1(A,A,A,C,E,G)),B) ) ) ) ) ) ) ) ) ) ).
fof(d3_filter_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,A)
& m2_relset_1(C,A,A) )
=> ( m1_filter_1(C,A,B)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k8_eqrel_1(A,B),k8_eqrel_1(A,B))
& m2_relset_1(D,k8_eqrel_1(A,B),k8_eqrel_1(A,B)) )
=> ( D = k3_filter_1(A,B,C)
<=> ! [E,F] :
( ( r2_hidden(E,k8_eqrel_1(A,B))
& r2_hidden(F,E) )
=> k1_funct_1(D,E) = k6_eqrel_1(A,B,k1_funct_1(C,F)) ) ) ) ) ) ) ) ).
fof(d4_filter_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ( m2_filter_1(C,A,B)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(k8_eqrel_1(A,B),k8_eqrel_1(A,B)),k8_eqrel_1(A,B))
& m2_relset_1(D,k2_zfmisc_1(k8_eqrel_1(A,B),k8_eqrel_1(A,B)),k8_eqrel_1(A,B)) )
=> ( D = k4_filter_1(A,B,C)
<=> ! [E,F,G,H] :
( ( r2_hidden(E,k8_eqrel_1(A,B))
& r2_hidden(F,k8_eqrel_1(A,B))
& r2_hidden(G,E)
& r2_hidden(H,F) )
=> k1_binop_1(D,E,F) = k6_eqrel_1(A,B,k1_binop_1(C,G,H)) ) ) ) ) ) ) ) ).
fof(t3_filter_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m2_filter_1(E,A,B)
=> k2_binop_1(k8_eqrel_1(A,B),k8_eqrel_1(A,B),k8_eqrel_1(A,B),k4_filter_1(A,B,E),k2_filter_1(A,B,C),k2_filter_1(A,B,D)) = k2_filter_1(A,B,k2_binop_1(A,A,A,E,C,D)) ) ) ) ) ) ).
fof(t4_filter_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C] :
( m2_filter_1(C,A,B)
=> ( v1_binop_1(C,A)
=> v1_binop_1(k4_filter_1(A,B,C),k8_eqrel_1(A,B)) ) ) ) ) ).
fof(t5_filter_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C] :
( m2_filter_1(C,A,B)
=> ( v2_binop_1(C,A)
=> v2_binop_1(k4_filter_1(A,B,C),k8_eqrel_1(A,B)) ) ) ) ) ).
fof(t6_filter_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m2_filter_1(D,A,B)
=> ( r1_binop_1(A,C,D)
=> r1_binop_1(k8_eqrel_1(A,B),k2_filter_1(A,B,C),k4_filter_1(A,B,D)) ) ) ) ) ) ).
fof(t7_filter_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m2_filter_1(D,A,B)
=> ( r2_binop_1(A,C,D)
=> r2_binop_1(k8_eqrel_1(A,B),k2_filter_1(A,B,C),k4_filter_1(A,B,D)) ) ) ) ) ) ).
fof(t8_filter_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m2_filter_1(D,A,B)
=> ( r3_binop_1(A,C,D)
=> r3_binop_1(k8_eqrel_1(A,B),k2_filter_1(A,B,C),k4_filter_1(A,B,D)) ) ) ) ) ) ).
fof(t9_filter_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C] :
( m2_filter_1(C,A,B)
=> ! [D] :
( m2_filter_1(D,A,B)
=> ( r4_binop_1(A,C,D)
=> r4_binop_1(k8_eqrel_1(A,B),k4_filter_1(A,B,C),k4_filter_1(A,B,D)) ) ) ) ) ) ).
fof(t10_filter_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C] :
( m2_filter_1(C,A,B)
=> ! [D] :
( m2_filter_1(D,A,B)
=> ( r5_binop_1(A,C,D)
=> r5_binop_1(k8_eqrel_1(A,B),k4_filter_1(A,B,C),k4_filter_1(A,B,D)) ) ) ) ) ) ).
fof(t11_filter_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C] :
( m2_filter_1(C,A,B)
=> ! [D] :
( m2_filter_1(D,A,B)
=> ( r6_binop_1(A,C,D)
=> r6_binop_1(k8_eqrel_1(A,B),k4_filter_1(A,B,C),k4_filter_1(A,B,D)) ) ) ) ) ) ).
fof(t12_filter_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C] :
( m2_filter_1(C,A,B)
=> ! [D] :
( m2_filter_1(D,A,B)
=> ( r1_lattice2(A,C,D)
=> r1_lattice2(k8_eqrel_1(A,B),k4_filter_1(A,B,C),k4_filter_1(A,B,D)) ) ) ) ) ) ).
fof(t13_filter_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v3_filter_0(A)
& l3_lattices(A) )
=> ! [B] :
( m1_filter_0(B,A)
=> m2_filter_1(u2_lattices(A),u1_struct_0(A),k11_filter_0(A,B)) ) ) ).
fof(t14_filter_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v3_filter_0(A)
& l3_lattices(A) )
=> ! [B] :
( m1_filter_0(B,A)
=> m2_filter_1(u1_lattices(A),u1_struct_0(A),k11_filter_0(A,B)) ) ) ).
fof(d5_filter_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_filter_0(B,A)
=> ( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v3_filter_0(A)
& l3_lattices(A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v3_lattices(C)
& v10_lattices(C)
& l3_lattices(C) )
=> ( C = k5_filter_1(A,B)
<=> ! [D] :
( ( v3_relat_2(D)
& v8_relat_2(D)
& v1_partfun1(D,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(D,u1_struct_0(A),u1_struct_0(A)) )
=> ( D = k10_filter_0(A,B)
=> C = g3_lattices(k8_eqrel_1(u1_struct_0(A),D),k4_filter_1(u1_struct_0(A),D,u2_lattices(A)),k4_filter_1(u1_struct_0(A),D,u1_lattices(A))) ) ) ) ) ) ) ) ).
fof(d6_filter_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_filter_0(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v3_filter_0(A)
& l3_lattices(A) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k5_filter_1(A,B)))
=> ( D = k6_filter_1(A,B,C)
<=> ! [E] :
( ( v3_relat_2(E)
& v8_relat_2(E)
& v1_partfun1(E,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(E,u1_struct_0(A),u1_struct_0(A)) )
=> ( E = k10_filter_0(A,B)
=> D = k2_filter_1(u1_struct_0(A),E,C) ) ) ) ) ) ) ) ) ).
fof(t15_filter_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v3_filter_0(A)
& l3_lattices(A) )
=> ! [B] :
( m1_filter_0(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( k3_lattices(k5_filter_1(A,B),k6_filter_1(A,B,C),k6_filter_1(A,B,D)) = k6_filter_1(A,B,k3_lattices(A,C,D))
& k4_lattices(k5_filter_1(A,B),k6_filter_1(A,B,C),k6_filter_1(A,B,D)) = k6_filter_1(A,B,k4_lattices(A,C,D)) ) ) ) ) ) ).
fof(t16_filter_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v3_filter_0(A)
& l3_lattices(A) )
=> ! [B] :
( m1_filter_0(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r3_lattices(k5_filter_1(A,B),k6_filter_1(A,B,C),k6_filter_1(A,B,D))
<=> r2_hidden(k4_filter_0(A,C,D),B) ) ) ) ) ) ).
fof(t17_filter_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v3_filter_0(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k4_filter_0(A,k4_lattices(A,B,C),D) = k4_filter_0(A,B,k4_filter_0(A,C,D)) ) ) ) ) ).
fof(t18_filter_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v3_filter_0(A)
& l3_lattices(A) )
=> ! [B] :
( m1_filter_0(B,A)
=> ( v13_lattices(A)
=> ( ~ v3_struct_0(k5_filter_1(A,B))
& v10_lattices(k5_filter_1(A,B))
& v13_lattices(k5_filter_1(A,B))
& l3_lattices(k5_filter_1(A,B))
& k5_lattices(k5_filter_1(A,B)) = k6_filter_1(A,B,k5_lattices(A)) ) ) ) ) ).
fof(t19_filter_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v3_filter_0(A)
& l3_lattices(A) )
=> ! [B] :
( m1_filter_0(B,A)
=> ( ~ v3_struct_0(k5_filter_1(A,B))
& v10_lattices(k5_filter_1(A,B))
& v14_lattices(k5_filter_1(A,B))
& l3_lattices(k5_filter_1(A,B))
& k6_lattices(k5_filter_1(A,B)) = k6_filter_1(A,B,k6_lattices(A)) ) ) ) ).
fof(t20_filter_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v3_filter_0(A)
& l3_lattices(A) )
=> ! [B] :
( m1_filter_0(B,A)
=> v3_filter_0(k5_filter_1(A,B)) ) ) ).
fof(t21_filter_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_filter_0(B,A)
=> ( ~ v3_struct_0(k5_filter_1(A,B))
& v10_lattices(k5_filter_1(A,B))
& v17_lattices(k5_filter_1(A,B))
& l3_lattices(k5_filter_1(A,B)) ) ) ) ).
fof(t22_filter_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,B)
=> ! [F] :
( m1_subset_1(F,B)
=> ! [G] :
( ( v1_funct_1(G)
& v1_funct_2(G,k2_zfmisc_1(A,A),A)
& m2_relset_1(G,k2_zfmisc_1(A,A),A) )
=> ! [H] :
( ( v1_funct_1(H)
& v1_funct_2(H,k2_zfmisc_1(B,B),B)
& m2_relset_1(H,k2_zfmisc_1(B,B),B) )
=> k2_binop_1(k2_zfmisc_1(A,B),k2_zfmisc_1(A,B),k2_zfmisc_1(A,B),k7_filter_1(A,B,G,H),k1_domain_1(A,B,C,E),k1_domain_1(A,B,D,F)) = k1_domain_1(A,B,k2_binop_1(A,A,A,G,C,D),k2_binop_1(B,B,B,H,E,F)) ) ) ) ) ) ) ) ) ).
fof(t23_filter_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(B,B),B)
& m2_relset_1(D,k2_zfmisc_1(B,B),B) )
=> ( ( v1_binop_1(C,A)
& v1_binop_1(D,B) )
<=> v1_binop_1(k7_filter_1(A,B,C,D),k2_zfmisc_1(A,B)) ) ) ) ) ) ).
fof(t24_filter_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(B,B),B)
& m2_relset_1(D,k2_zfmisc_1(B,B),B) )
=> ( ( v2_binop_1(C,A)
& v2_binop_1(D,B) )
<=> v2_binop_1(k7_filter_1(A,B,C,D),k2_zfmisc_1(A,B)) ) ) ) ) ) ).
fof(t25_filter_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,B)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k2_zfmisc_1(A,A),A)
& m2_relset_1(E,k2_zfmisc_1(A,A),A) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k2_zfmisc_1(B,B),B)
& m2_relset_1(F,k2_zfmisc_1(B,B),B) )
=> ( ( r1_binop_1(A,C,E)
& r1_binop_1(B,D,F) )
<=> r1_binop_1(k2_zfmisc_1(A,B),k1_domain_1(A,B,C,D),k7_filter_1(A,B,E,F)) ) ) ) ) ) ) ) ).
fof(t26_filter_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,B)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k2_zfmisc_1(A,A),A)
& m2_relset_1(E,k2_zfmisc_1(A,A),A) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k2_zfmisc_1(B,B),B)
& m2_relset_1(F,k2_zfmisc_1(B,B),B) )
=> ( ( r2_binop_1(A,C,E)
& r2_binop_1(B,D,F) )
<=> r2_binop_1(k2_zfmisc_1(A,B),k1_domain_1(A,B,C,D),k7_filter_1(A,B,E,F)) ) ) ) ) ) ) ) ).
fof(t27_filter_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,B)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k2_zfmisc_1(A,A),A)
& m2_relset_1(E,k2_zfmisc_1(A,A),A) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k2_zfmisc_1(B,B),B)
& m2_relset_1(F,k2_zfmisc_1(B,B),B) )
=> ( ( r3_binop_1(A,C,E)
& r3_binop_1(B,D,F) )
<=> r3_binop_1(k2_zfmisc_1(A,B),k1_domain_1(A,B,C,D),k7_filter_1(A,B,E,F)) ) ) ) ) ) ) ) ).
fof(t28_filter_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(A,A),A)
& m2_relset_1(D,k2_zfmisc_1(A,A),A) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k2_zfmisc_1(B,B),B)
& m2_relset_1(E,k2_zfmisc_1(B,B),B) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k2_zfmisc_1(B,B),B)
& m2_relset_1(F,k2_zfmisc_1(B,B),B) )
=> ( ( r4_binop_1(A,C,D)
& r4_binop_1(B,E,F) )
<=> r4_binop_1(k2_zfmisc_1(A,B),k7_filter_1(A,B,C,E),k7_filter_1(A,B,D,F)) ) ) ) ) ) ) ) ).
fof(t29_filter_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(A,A),A)
& m2_relset_1(D,k2_zfmisc_1(A,A),A) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k2_zfmisc_1(B,B),B)
& m2_relset_1(E,k2_zfmisc_1(B,B),B) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k2_zfmisc_1(B,B),B)
& m2_relset_1(F,k2_zfmisc_1(B,B),B) )
=> ( ( r5_binop_1(A,C,D)
& r5_binop_1(B,E,F) )
<=> r5_binop_1(k2_zfmisc_1(A,B),k7_filter_1(A,B,C,E),k7_filter_1(A,B,D,F)) ) ) ) ) ) ) ) ).
fof(t30_filter_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(A,A),A)
& m2_relset_1(D,k2_zfmisc_1(A,A),A) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k2_zfmisc_1(B,B),B)
& m2_relset_1(E,k2_zfmisc_1(B,B),B) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k2_zfmisc_1(B,B),B)
& m2_relset_1(F,k2_zfmisc_1(B,B),B) )
=> ( ( r6_binop_1(A,C,D)
& r6_binop_1(B,E,F) )
<=> r6_binop_1(k2_zfmisc_1(A,B),k7_filter_1(A,B,C,E),k7_filter_1(A,B,D,F)) ) ) ) ) ) ) ) ).
fof(t31_filter_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(A,A),A)
& m2_relset_1(D,k2_zfmisc_1(A,A),A) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k2_zfmisc_1(B,B),B)
& m2_relset_1(E,k2_zfmisc_1(B,B),B) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k2_zfmisc_1(B,B),B)
& m2_relset_1(F,k2_zfmisc_1(B,B),B) )
=> ( ( r1_lattice2(A,C,D)
& r1_lattice2(B,E,F) )
<=> r1_lattice2(k2_zfmisc_1(A,B),k7_filter_1(A,B,C,E),k7_filter_1(A,B,D,F)) ) ) ) ) ) ) ) ).
fof(d7_filter_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l3_lattices(B) )
=> k8_filter_1(A,B) = g3_lattices(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k7_filter_1(u1_struct_0(A),u1_struct_0(B),u2_lattices(A),u2_lattices(B)),k7_filter_1(u1_struct_0(A),u1_struct_0(B),u1_lattices(A),u1_lattices(B))) ) ) ).
fof(t32_filter_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r2_hidden(k1_domain_1(u1_struct_0(A),u1_struct_0(A),B,C),k9_filter_1(A))
<=> r3_lattices(A,B,C) ) ) ) ) ).
fof(t33_filter_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ( k1_relat_1(k9_filter_1(A)) = u1_struct_0(A)
& k2_relat_1(k9_filter_1(A)) = u1_struct_0(A)
& k3_relat_1(k9_filter_1(A)) = u1_struct_0(A) ) ) ).
fof(d9_filter_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& l3_lattices(B) )
=> ( r1_filter_1(A,B)
<=> r4_wellord1(k9_filter_1(A),k9_filter_1(B)) ) ) ) ).
fof(t34_filter_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& l3_lattices(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v10_lattices(C)
& l3_lattices(C) )
=> ( ( r1_filter_1(A,B)
& r1_filter_1(B,C) )
=> r1_filter_1(A,C) ) ) ) ) ).
fof(t35_filter_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l3_lattices(B) )
=> ( ( ~ v3_struct_0(k8_filter_1(A,B))
& v10_lattices(k8_filter_1(A,B))
& l3_lattices(k8_filter_1(A,B)) )
=> ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A)
& ~ v3_struct_0(B)
& v10_lattices(B)
& l3_lattices(B) ) ) ) ) ).
fof(t36_filter_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& l3_lattices(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(B))
=> ( k3_lattices(k8_filter_1(A,B),k10_filter_1(A,B,C,E),k10_filter_1(A,B,D,F)) = k10_filter_1(A,B,k3_lattices(A,C,D),k3_lattices(B,E,F))
& k4_lattices(k8_filter_1(A,B),k10_filter_1(A,B,C,E),k10_filter_1(A,B,D,F)) = k10_filter_1(A,B,k4_lattices(A,C,D),k4_lattices(B,E,F)) ) ) ) ) ) ) ) ).
fof(t37_filter_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& l3_lattices(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(B))
=> ( r3_lattices(k8_filter_1(A,B),k10_filter_1(A,B,C,E),k10_filter_1(A,B,D,F))
<=> ( r3_lattices(A,C,D)
& r3_lattices(B,E,F) ) ) ) ) ) ) ) ) ).
fof(t38_filter_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& l3_lattices(B) )
=> ( ( v12_lattices(A)
& v12_lattices(B) )
<=> v12_lattices(k8_filter_1(A,B)) ) ) ) ).
fof(t39_filter_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& l3_lattices(B) )
=> ( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A)
& l3_lattices(A)
& ~ v3_struct_0(B)
& v10_lattices(B)
& v11_lattices(B)
& l3_lattices(B) )
<=> ( ~ v3_struct_0(k8_filter_1(A,B))
& v10_lattices(k8_filter_1(A,B))
& v11_lattices(k8_filter_1(A,B))
& l3_lattices(k8_filter_1(A,B)) ) ) ) ) ).
fof(t40_filter_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& l3_lattices(B) )
=> ( ( v13_lattices(A)
& v13_lattices(B) )
<=> v13_lattices(k8_filter_1(A,B)) ) ) ) ).
fof(t41_filter_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& l3_lattices(B) )
=> ( ( v14_lattices(A)
& v14_lattices(B) )
<=> v14_lattices(k8_filter_1(A,B)) ) ) ) ).
fof(t42_filter_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& l3_lattices(B) )
=> ( ( v15_lattices(A)
& v15_lattices(B) )
<=> v15_lattices(k8_filter_1(A,B)) ) ) ) ).
fof(t43_filter_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& l3_lattices(B) )
=> ( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v13_lattices(A)
& l3_lattices(A)
& ~ v3_struct_0(B)
& v10_lattices(B)
& v13_lattices(B)
& l3_lattices(B) )
=> k5_lattices(k8_filter_1(A,B)) = k10_filter_1(A,B,k5_lattices(A),k5_lattices(B)) ) ) ) ).
fof(t44_filter_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& l3_lattices(B) )
=> ( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v14_lattices(A)
& l3_lattices(A)
& ~ v3_struct_0(B)
& v10_lattices(B)
& v14_lattices(B)
& l3_lattices(B) )
=> k6_lattices(k8_filter_1(A,B)) = k10_filter_1(A,B,k6_lattices(A),k6_lattices(B)) ) ) ) ).
fof(t45_filter_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& l3_lattices(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(B))
=> ( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v15_lattices(A)
& l3_lattices(A)
& ~ v3_struct_0(B)
& v10_lattices(B)
& v15_lattices(B)
& l3_lattices(B) )
=> ( ( r2_lattices(A,C,D)
& r2_lattices(B,E,F) )
<=> r2_lattices(k8_filter_1(A,B),k10_filter_1(A,B,C,E),k10_filter_1(A,B,D,F)) ) ) ) ) ) ) ) ) ).
fof(t46_filter_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& l3_lattices(B) )
=> ( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v15_lattices(A)
& v16_lattices(A)
& l3_lattices(A)
& ~ v3_struct_0(B)
& v10_lattices(B)
& v15_lattices(B)
& v16_lattices(B)
& l3_lattices(B) )
<=> ( ~ v3_struct_0(k8_filter_1(A,B))
& v10_lattices(k8_filter_1(A,B))
& v15_lattices(k8_filter_1(A,B))
& v16_lattices(k8_filter_1(A,B))
& l3_lattices(k8_filter_1(A,B)) ) ) ) ) ).
fof(t47_filter_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& l3_lattices(B) )
=> ( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& l3_lattices(A)
& ~ v3_struct_0(B)
& v10_lattices(B)
& v17_lattices(B)
& l3_lattices(B) )
<=> ( ~ v3_struct_0(k8_filter_1(A,B))
& v10_lattices(k8_filter_1(A,B))
& v17_lattices(k8_filter_1(A,B))
& l3_lattices(k8_filter_1(A,B)) ) ) ) ) ).
fof(t48_filter_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& l3_lattices(B) )
=> ( ( v3_filter_0(A)
& v3_filter_0(B) )
<=> v3_filter_0(k8_filter_1(A,B)) ) ) ) ).
fof(t49_filter_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& l3_lattices(B) )
=> k1_lattice2(k8_filter_1(A,B)) = k8_filter_1(k1_lattice2(A),k1_lattice2(B)) ) ) ).
fof(t50_filter_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& l3_lattices(B) )
=> r1_filter_1(k8_filter_1(A,B),k8_filter_1(B,A)) ) ) ).
fof(t51_filter_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k9_filter_0(A,B,C) = k3_lattices(A,k4_lattices(A,B,C),k4_lattices(A,k7_lattices(A,B),k7_lattices(A,C))) ) ) ) ).
fof(t52_filter_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( k7_lattices(A,k4_filter_0(A,B,C)) = k4_lattices(A,B,k7_lattices(A,C))
& k7_lattices(A,k9_filter_0(A,B,C)) = k3_lattices(A,k4_lattices(A,B,k7_lattices(A,C)),k4_lattices(A,k7_lattices(A,B),C))
& k7_lattices(A,k9_filter_0(A,B,C)) = k9_filter_0(A,B,k7_lattices(A,C))
& k7_lattices(A,k9_filter_0(A,B,C)) = k9_filter_0(A,k7_lattices(A,B),C) ) ) ) ) ).
fof(t53_filter_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( k9_filter_0(A,B,C) = k9_filter_0(A,B,D)
=> C = D ) ) ) ) ) ).
fof(t54_filter_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k9_filter_0(A,B,k9_filter_0(A,B,C)) = C ) ) ) ).
fof(t55_filter_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v3_filter_0(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( k4_filter_0(A,k3_lattices(A,B,C),B) = k4_filter_0(A,C,B)
& k4_filter_0(A,B,k4_lattices(A,B,C)) = k4_filter_0(A,B,C) ) ) ) ) ).
fof(t56_filter_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v3_filter_0(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r3_lattices(A,k4_filter_0(A,B,C),k4_filter_0(A,B,k3_lattices(A,C,D)))
& r3_lattices(A,k4_filter_0(A,B,C),k4_filter_0(A,k4_lattices(A,B,D),C))
& r3_lattices(A,k4_filter_0(A,B,C),k4_filter_0(A,B,k3_lattices(A,D,C)))
& r3_lattices(A,k4_filter_0(A,B,C),k4_filter_0(A,k4_lattices(A,D,B),C)) ) ) ) ) ) ).
fof(t57_filter_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v3_filter_0(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> r3_lattices(A,k4_lattices(A,k4_filter_0(A,B,C),k4_filter_0(A,D,C)),k4_filter_0(A,k3_lattices(A,B,D),C)) ) ) ) ) ).
fof(t58_filter_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v3_filter_0(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> r3_lattices(A,k4_lattices(A,k4_filter_0(A,B,C),k4_filter_0(A,B,D)),k4_filter_0(A,B,k4_lattices(A,C,D))) ) ) ) ) ).
fof(t59_filter_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v3_filter_0(A)
& l3_lattices(A) )
=> ! [B] :
( m1_filter_0(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ( ( r2_hidden(k9_filter_0(A,C,D),B)
& r2_hidden(k9_filter_0(A,E,F),B) )
=> ( r2_hidden(k9_filter_0(A,k3_lattices(A,C,E),k3_lattices(A,D,F)),B)
& r2_hidden(k9_filter_0(A,k4_lattices(A,C,E),k4_lattices(A,D,F)),B) ) ) ) ) ) ) ) ) ).
fof(t60_filter_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v3_filter_0(A)
& l3_lattices(A) )
=> ! [B] :
( m1_filter_0(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( ( r2_hidden(C,k2_filter_1(u1_struct_0(A),k11_filter_0(A,B),D))
& r2_hidden(E,k2_filter_1(u1_struct_0(A),k11_filter_0(A,B),D)) )
=> ( r2_hidden(k3_lattices(A,C,E),k2_filter_1(u1_struct_0(A),k11_filter_0(A,B),D))
& r2_hidden(k4_lattices(A,C,E),k2_filter_1(u1_struct_0(A),k11_filter_0(A,B),D)) ) ) ) ) ) ) ) ).
fof(t61_filter_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r2_hidden(k3_lattices(A,B,k9_filter_0(A,B,C)),k2_filter_1(u1_struct_0(A),k12_filter_0(A,k2_filter_0(A,C)),B))
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r2_hidden(D,k2_filter_1(u1_struct_0(A),k12_filter_0(A,k2_filter_0(A,C)),B))
=> r3_lattices(A,D,k3_lattices(A,B,k9_filter_0(A,B,C))) ) ) ) ) ) ) ).
fof(t62_filter_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> r1_filter_1(A,k8_filter_1(k5_filter_1(A,k2_filter_0(A,B)),k8_filter_0(A,k2_filter_0(A,B)))) ) ) ).
fof(s1_filter_1,axiom,
( ! [A] :
( m1_subset_1(A,f1_s1_filter_1)
=> p1_s1_filter_1(k2_filter_1(f1_s1_filter_1,f2_s1_filter_1,A)) )
=> ! [A] :
( m2_subset_1(A,k1_zfmisc_1(f1_s1_filter_1),k8_eqrel_1(f1_s1_filter_1,f2_s1_filter_1))
=> p1_s1_filter_1(A) ) ) ).
fof(s2_filter_1,axiom,
( ! [A] :
( m1_subset_1(A,f1_s2_filter_1)
=> ! [B] :
( m1_subset_1(B,f1_s2_filter_1)
=> p1_s2_filter_1(k2_filter_1(f1_s2_filter_1,f2_s2_filter_1,A),k2_filter_1(f1_s2_filter_1,f2_s2_filter_1,B)) ) )
=> ! [A] :
( m2_subset_1(A,k1_zfmisc_1(f1_s2_filter_1),k8_eqrel_1(f1_s2_filter_1,f2_s2_filter_1))
=> ! [B] :
( m2_subset_1(B,k1_zfmisc_1(f1_s2_filter_1),k8_eqrel_1(f1_s2_filter_1,f2_s2_filter_1))
=> p1_s2_filter_1(A,B) ) ) ) ).
fof(s3_filter_1,axiom,
( ! [A] :
( m1_subset_1(A,f1_s3_filter_1)
=> ! [B] :
( m1_subset_1(B,f1_s3_filter_1)
=> ! [C] :
( m1_subset_1(C,f1_s3_filter_1)
=> p1_s3_filter_1(k2_filter_1(f1_s3_filter_1,f2_s3_filter_1,A),k2_filter_1(f1_s3_filter_1,f2_s3_filter_1,B),k2_filter_1(f1_s3_filter_1,f2_s3_filter_1,C)) ) ) )
=> ! [A] :
( m2_subset_1(A,k1_zfmisc_1(f1_s3_filter_1),k8_eqrel_1(f1_s3_filter_1,f2_s3_filter_1))
=> ! [B] :
( m2_subset_1(B,k1_zfmisc_1(f1_s3_filter_1),k8_eqrel_1(f1_s3_filter_1,f2_s3_filter_1))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(f1_s3_filter_1),k8_eqrel_1(f1_s3_filter_1,f2_s3_filter_1))
=> p1_s3_filter_1(A,B,C) ) ) ) ) ).
fof(s4_filter_1,axiom,
( ! [A] :
( m1_subset_1(A,f1_s4_filter_1)
=> ! [B] :
( m1_subset_1(B,f2_s4_filter_1)
=> p1_s4_filter_1(k1_domain_1(f1_s4_filter_1,f2_s4_filter_1,A,B)) ) )
=> ! [A] :
( m1_subset_1(A,k2_zfmisc_1(f1_s4_filter_1,f2_s4_filter_1))
=> p1_s4_filter_1(A) ) ) ).
fof(s5_filter_1,axiom,
( ! [A] :
( m1_subset_1(A,f1_s5_filter_1)
=> ! [B] :
( m1_subset_1(B,f1_s5_filter_1)
=> ! [C] :
( m1_subset_1(C,f2_s5_filter_1)
=> ! [D] :
( m1_subset_1(D,f2_s5_filter_1)
=> p1_s5_filter_1(k1_domain_1(f1_s5_filter_1,f2_s5_filter_1,A,C),k1_domain_1(f1_s5_filter_1,f2_s5_filter_1,B,D)) ) ) ) )
=> ! [A] :
( m1_subset_1(A,k2_zfmisc_1(f1_s5_filter_1,f2_s5_filter_1))
=> ! [B] :
( m1_subset_1(B,k2_zfmisc_1(f1_s5_filter_1,f2_s5_filter_1))
=> p1_s5_filter_1(A,B) ) ) ) ).
fof(s6_filter_1,axiom,
( ! [A] :
( m1_subset_1(A,f1_s6_filter_1)
=> ! [B] :
( m1_subset_1(B,f1_s6_filter_1)
=> ! [C] :
( m1_subset_1(C,f1_s6_filter_1)
=> ! [D] :
( m1_subset_1(D,f2_s6_filter_1)
=> ! [E] :
( m1_subset_1(E,f2_s6_filter_1)
=> ! [F] :
( m1_subset_1(F,f2_s6_filter_1)
=> p1_s6_filter_1(k1_domain_1(f1_s6_filter_1,f2_s6_filter_1,A,D),k1_domain_1(f1_s6_filter_1,f2_s6_filter_1,B,E),k1_domain_1(f1_s6_filter_1,f2_s6_filter_1,C,F)) ) ) ) ) ) )
=> ! [A] :
( m1_subset_1(A,k2_zfmisc_1(f1_s6_filter_1,f2_s6_filter_1))
=> ! [B] :
( m1_subset_1(B,k2_zfmisc_1(f1_s6_filter_1,f2_s6_filter_1))
=> ! [C] :
( m1_subset_1(C,k2_zfmisc_1(f1_s6_filter_1,f2_s6_filter_1))
=> p1_s6_filter_1(A,B,C) ) ) ) ) ).
fof(dt_m1_filter_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_relat_1(B) )
=> ! [C] :
( m1_filter_1(C,A,B)
=> ( v1_funct_1(C)
& v1_funct_2(C,A,A)
& m2_relset_1(C,A,A) ) ) ) ).
fof(existence_m1_filter_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_relat_1(B) )
=> ? [C] : m1_filter_1(C,A,B) ) ).
fof(dt_m2_filter_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_relat_1(B) )
=> ! [C] :
( m2_filter_1(C,A,B)
=> ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) ) ) ) ).
fof(existence_m2_filter_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_relat_1(B) )
=> ? [C] : m2_filter_1(C,A,B) ) ).
fof(symmetry_r1_filter_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A)
& ~ v3_struct_0(B)
& v10_lattices(B)
& l3_lattices(B) )
=> ( r1_filter_1(A,B)
=> r1_filter_1(B,A) ) ) ).
fof(reflexivity_r1_filter_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A)
& ~ v3_struct_0(B)
& v10_lattices(B)
& l3_lattices(B) )
=> r1_filter_1(A,A) ) ).
fof(dt_k1_filter_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A)
& m1_filter_0(B,A)
& m1_filter_0(C,A) )
=> m1_filter_0(k1_filter_1(A,B,C),A) ) ).
fof(commutativity_k1_filter_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A)
& m1_filter_0(B,A)
& m1_filter_0(C,A) )
=> k1_filter_1(A,B,C) = k1_filter_1(A,C,B) ) ).
fof(idempotence_k1_filter_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A)
& m1_filter_0(B,A)
& m1_filter_0(C,A) )
=> k1_filter_1(A,B,B) = B ) ).
fof(redefinition_k1_filter_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A)
& m1_filter_0(B,A)
& m1_filter_0(C,A) )
=> k1_filter_1(A,B,C) = k3_xboole_0(B,C) ) ).
fof(dt_k2_filter_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m1_relset_1(B,A,A)
& m1_subset_1(C,A) )
=> m2_subset_1(k2_filter_1(A,B,C),k1_zfmisc_1(A),k8_eqrel_1(A,B)) ) ).
fof(redefinition_k2_filter_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m1_relset_1(B,A,A)
& m1_subset_1(C,A) )
=> k2_filter_1(A,B,C) = k6_eqrel_1(A,B,C) ) ).
fof(dt_k3_filter_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m1_relset_1(B,A,A)
& v1_funct_1(C)
& v1_funct_2(C,A,A)
& m1_relset_1(C,A,A) )
=> ( v1_funct_1(k3_filter_1(A,B,C))
& v1_funct_2(k3_filter_1(A,B,C),k8_eqrel_1(A,B),k8_eqrel_1(A,B))
& m2_relset_1(k3_filter_1(A,B,C),k8_eqrel_1(A,B),k8_eqrel_1(A,B)) ) ) ).
fof(dt_k4_filter_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m1_relset_1(B,A,A)
& v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m1_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ( v1_funct_1(k4_filter_1(A,B,C))
& v1_funct_2(k4_filter_1(A,B,C),k2_zfmisc_1(k8_eqrel_1(A,B),k8_eqrel_1(A,B)),k8_eqrel_1(A,B))
& m2_relset_1(k4_filter_1(A,B,C),k2_zfmisc_1(k8_eqrel_1(A,B),k8_eqrel_1(A,B)),k8_eqrel_1(A,B)) ) ) ).
fof(dt_k5_filter_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A)
& m1_filter_0(B,A) )
=> ( ~ v3_struct_0(k5_filter_1(A,B))
& v3_lattices(k5_filter_1(A,B))
& v10_lattices(k5_filter_1(A,B))
& l3_lattices(k5_filter_1(A,B)) ) ) ).
fof(dt_k6_filter_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A)
& m1_filter_0(B,A)
& m1_subset_1(C,u1_struct_0(A)) )
=> m1_subset_1(k6_filter_1(A,B,C),u1_struct_0(k5_filter_1(A,B))) ) ).
fof(dt_k7_filter_1,axiom,
! [A,B,C,D] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m1_relset_1(C,k2_zfmisc_1(A,A),A)
& v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(B,B),B)
& m1_relset_1(D,k2_zfmisc_1(B,B),B) )
=> ( v1_funct_1(k7_filter_1(A,B,C,D))
& v1_funct_2(k7_filter_1(A,B,C,D),k2_zfmisc_1(k2_zfmisc_1(A,B),k2_zfmisc_1(A,B)),k2_zfmisc_1(A,B))
& m2_relset_1(k7_filter_1(A,B,C,D),k2_zfmisc_1(k2_zfmisc_1(A,B),k2_zfmisc_1(A,B)),k2_zfmisc_1(A,B)) ) ) ).
fof(redefinition_k7_filter_1,axiom,
! [A,B,C,D] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m1_relset_1(C,k2_zfmisc_1(A,A),A)
& v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(B,B),B)
& m1_relset_1(D,k2_zfmisc_1(B,B),B) )
=> k7_filter_1(A,B,C,D) = k3_funct_4(C,D) ) ).
fof(dt_k8_filter_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l3_lattices(A)
& ~ v3_struct_0(B)
& l3_lattices(B) )
=> ( v3_lattices(k8_filter_1(A,B))
& l3_lattices(k8_filter_1(A,B)) ) ) ).
fof(dt_k9_filter_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> v1_relat_1(k9_filter_1(A)) ) ).
fof(dt_k10_filter_1,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A)
& ~ v3_struct_0(B)
& v10_lattices(B)
& l3_lattices(B)
& m1_subset_1(C,u1_struct_0(A))
& m1_subset_1(D,u1_struct_0(B)) )
=> m1_subset_1(k10_filter_1(A,B,C,D),u1_struct_0(k8_filter_1(A,B))) ) ).
fof(redefinition_k10_filter_1,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A)
& ~ v3_struct_0(B)
& v10_lattices(B)
& l3_lattices(B)
& m1_subset_1(C,u1_struct_0(A))
& m1_subset_1(D,u1_struct_0(B)) )
=> k10_filter_1(A,B,C,D) = k4_tarski(C,D) ) ).
fof(d8_filter_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> k9_filter_1(A) = a_1_0_filter_1(A) ) ).
fof(fraenkel_a_1_0_filter_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& l3_lattices(B) )
=> ( r2_hidden(A,a_1_0_filter_1(B))
<=> ? [C,D] :
( m1_subset_1(C,u1_struct_0(B))
& m1_subset_1(D,u1_struct_0(B))
& A = k1_domain_1(u1_struct_0(B),u1_struct_0(B),C,D)
& r3_lattices(B,C,D) ) ) ) ).
%------------------------------------------------------------------------------