SET007 Axioms: SET007+482.ax
%------------------------------------------------------------------------------
% File : SET007+482 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Fixpoints in Complete Lattices
% 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 : knaster [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 88 ( 6 unt; 0 def)
% Number of atoms : 912 ( 64 equ)
% Maximal formula atoms : 23 ( 10 avg)
% Number of connectives : 923 ( 99 ~; 4 |; 529 &)
% ( 16 <=>; 275 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 11 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 48 ( 46 usr; 1 prp; 0-3 aty)
% Number of functors : 44 ( 44 usr; 4 con; 0-4 aty)
% Number of variables : 303 ( 280 !; 23 ?)
% 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_knaster,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ? [C] :
( m1_relset_1(C,A,B)
& v1_relat_1(C)
& v1_funct_1(C)
& v1_funct_2(C,A,B)
& v6_cohsp_1(C) ) ) ).
fof(rc2_knaster,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& ~ v1_xboole_0(B)
& v2_knaster(B,A)
& v3_knaster(B,A) ) ) ).
fof(cc1_knaster,axiom,
! [A] :
( l3_lattices(A)
=> ( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A) )
=> ( ~ v3_struct_0(A)
& v4_lattices(A)
& v5_lattices(A)
& v6_lattices(A)
& v7_lattices(A)
& v8_lattices(A)
& v9_lattices(A)
& v10_lattices(A)
& v13_lattices(A)
& v14_lattices(A)
& v15_lattices(A) ) ) ) ).
fof(fc1_knaster,axiom,
! [A] :
( ~ v3_struct_0(k1_lattice3(A))
& v3_lattices(k1_lattice3(A))
& v4_lattices(k1_lattice3(A))
& v5_lattices(k1_lattice3(A))
& v6_lattices(k1_lattice3(A))
& v7_lattices(k1_lattice3(A))
& v8_lattices(k1_lattice3(A))
& v9_lattices(k1_lattice3(A))
& v10_lattices(k1_lattice3(A))
& v11_lattices(k1_lattice3(A))
& v12_lattices(k1_lattice3(A))
& v13_lattices(k1_lattice3(A))
& v14_lattices(k1_lattice3(A))
& v15_lattices(k1_lattice3(A))
& v16_lattices(k1_lattice3(A))
& v17_lattices(k1_lattice3(A))
& v4_lattice3(k1_lattice3(A)) ) ).
fof(t1_knaster,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( ( v2_funct_1(A)
& v2_funct_1(B)
& r1_xboole_0(k2_relat_1(A),k2_relat_1(B)) )
=> v2_funct_1(k1_funct_4(A,B)) ) ) ) ).
fof(t2_knaster,axiom,
$true ).
fof(t3_knaster,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( ( A = k2_xboole_0(B,C)
& r1_xboole_0(k1_relat_1(B),k1_relat_1(C)) )
=> ( v2_funct_1(A)
<=> ( v2_funct_1(B)
& v2_funct_1(C)
& r1_xboole_0(k2_relat_1(B),k2_relat_1(C)) ) ) ) ) ) ) ).
fof(t4_knaster,axiom,
! [A,B,C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,A)
& m2_relset_1(D,A,A) )
=> ( r1_abian(B,k1_abian(A,D,C))
=> r1_abian(k1_funct_1(D,B),k1_abian(A,D,C)) ) ) ) ).
fof(t5_knaster,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,A)
& m2_relset_1(C,A,A) )
=> ( ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& r1_abian(B,k1_abian(A,C,D))
& ! [E] :
( r1_abian(E,k1_abian(A,C,D))
=> B = E ) )
=> r1_abian(B,C) ) ) ).
fof(d1_knaster,axiom,
$true ).
fof(d2_knaster,axiom,
$true ).
fof(d3_knaster,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,B)
& m2_relset_1(C,A,B) )
=> ( v6_cohsp_1(C)
<=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,A)
=> ( r1_tarski(D,E)
=> r1_tarski(k8_funct_2(A,B,C,D),k8_funct_2(A,B,C,E)) ) ) ) ) ) ) ) ).
fof(t6_knaster,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
& v6_cohsp_1(B)
& m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A)) )
=> r2_abian(k1_zfmisc_1(A),k1_knaster(A,B),B) ) ).
fof(t7_knaster,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
& v6_cohsp_1(B)
& m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A)) )
=> r2_abian(k1_zfmisc_1(A),k2_knaster(A,B),B) ) ).
fof(t8_knaster,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
& v6_cohsp_1(B)
& m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A)) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( r1_tarski(k8_funct_2(k1_zfmisc_1(A),k1_zfmisc_1(A),B,C),C)
=> r1_tarski(k1_knaster(A,B),C) ) ) ) ).
fof(t9_knaster,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
& v6_cohsp_1(B)
& m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A)) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( r1_tarski(C,k8_funct_2(k1_zfmisc_1(A),k1_zfmisc_1(A),B,C))
=> r1_tarski(C,k2_knaster(A,B)) ) ) ) ).
fof(t10_knaster,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
& v6_cohsp_1(B)
& m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A)) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( r2_abian(k1_zfmisc_1(A),C,B)
=> ( r1_tarski(k1_knaster(A,B),C)
& r1_tarski(C,k2_knaster(A,B)) ) ) ) ) ).
fof(t11_knaster,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,B)
& m2_relset_1(C,A,B) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,B,A)
& m2_relset_1(D,B,A) )
=> ? [E,F,G,H] :
( r1_xboole_0(E,F)
& r1_xboole_0(G,H)
& k2_xboole_0(E,F) = A
& k2_xboole_0(G,H) = B
& k2_funct_2(A,B,C,E) = G
& k2_funct_2(B,A,D,H) = F ) ) ) ) ) ).
fof(t12_knaster,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,B)
& m2_relset_1(C,A,B) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,B,A)
& m2_relset_1(D,B,A) )
=> ~ ( v2_funct_1(C)
& v2_funct_1(D)
& ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,A,B)
& m2_relset_1(E,A,B) )
=> ~ v3_funct_2(E,A,B) ) ) ) ) ) ) ).
fof(t13_knaster,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ( ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,B,A)
& m2_relset_1(C,B,A)
& v3_funct_2(C,B,A) )
=> r2_tarski(B,A) ) ) ) ).
fof(t14_knaster,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,B)
& m2_relset_1(C,A,B) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,B,A)
& m2_relset_1(D,B,A) )
=> ( ( v2_funct_1(C)
& v2_funct_1(D) )
=> r2_tarski(A,B) ) ) ) ) ) ).
fof(d6_knaster,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( v3_ordinal1(D)
=> ! [E] :
( E = k4_knaster(A,B,C,D)
<=> ? [F] :
( v1_relat_1(F)
& v1_funct_1(F)
& v5_ordinal1(F)
& E = k1_ordinal2(F)
& k1_relat_1(F) = k1_ordinal1(D)
& k1_funct_1(F,k1_xboole_0) = C
& ! [G] :
( v3_ordinal1(G)
=> ( r2_hidden(k1_ordinal1(G),k1_ordinal1(D))
=> k1_funct_1(F,k1_ordinal1(G)) = k1_funct_1(B,k1_funct_1(F,G)) ) )
& ! [G] :
( v3_ordinal1(G)
=> ( ( r2_hidden(G,k1_ordinal1(D))
& v4_ordinal1(G) )
=> ( G = k1_xboole_0
| k1_funct_1(F,G) = k15_lattice3(A,k2_relat_1(k2_ordinal1(F,G))) ) ) ) ) ) ) ) ) ) ).
fof(d7_knaster,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( v3_ordinal1(D)
=> ! [E] :
( E = k5_knaster(A,B,C,D)
<=> ? [F] :
( v1_relat_1(F)
& v1_funct_1(F)
& v5_ordinal1(F)
& E = k1_ordinal2(F)
& k1_relat_1(F) = k1_ordinal1(D)
& k1_funct_1(F,k1_xboole_0) = C
& ! [G] :
( v3_ordinal1(G)
=> ( r2_hidden(k1_ordinal1(G),k1_ordinal1(D))
=> k1_funct_1(F,k1_ordinal1(G)) = k1_funct_1(B,k1_funct_1(F,G)) ) )
& ! [G] :
( v3_ordinal1(G)
=> ( ( r2_hidden(G,k1_ordinal1(D))
& v4_ordinal1(G) )
=> ( G = k1_xboole_0
| k1_funct_1(F,G) = k16_lattice3(A,k2_relat_1(k2_ordinal1(F,G))) ) ) ) ) ) ) ) ) ) ).
fof(t15_knaster,axiom,
$true ).
fof(t16_knaster,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k4_knaster(A,B,C,k1_xboole_0) = C ) ) ) ).
fof(t17_knaster,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k5_knaster(A,B,C,k1_xboole_0) = C ) ) ) ).
fof(t18_knaster,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( v3_ordinal1(D)
=> k4_knaster(A,B,C,k1_ordinal1(D)) = k1_funct_1(B,k4_knaster(A,B,C,D)) ) ) ) ) ).
fof(t19_knaster,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( v3_ordinal1(D)
=> k5_knaster(A,B,C,k1_ordinal1(D)) = k1_funct_1(B,k5_knaster(A,B,C,D)) ) ) ) ) ).
fof(t20_knaster,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( v3_ordinal1(D)
=> ! [E] :
( ( v1_relat_1(E)
& v1_funct_1(E)
& v5_ordinal1(E) )
=> ( ( v4_ordinal1(D)
& k1_relat_1(E) = D
& ! [F] :
( v3_ordinal1(F)
=> ( r2_hidden(F,D)
=> k1_funct_1(E,F) = k4_knaster(A,B,C,F) ) ) )
=> ( D = k1_xboole_0
| k4_knaster(A,B,C,D) = k15_lattice3(A,k2_relat_1(E)) ) ) ) ) ) ) ) ).
fof(t21_knaster,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( v3_ordinal1(D)
=> ! [E] :
( ( v1_relat_1(E)
& v1_funct_1(E)
& v5_ordinal1(E) )
=> ( ( v4_ordinal1(D)
& k1_relat_1(E) = D
& ! [F] :
( v3_ordinal1(F)
=> ( r2_hidden(F,D)
=> k1_funct_1(E,F) = k5_knaster(A,B,C,F) ) ) )
=> ( D = k1_xboole_0
| k5_knaster(A,B,C,D) = k16_lattice3(A,k2_relat_1(E)) ) ) ) ) ) ) ) ).
fof(t22_knaster,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& l3_lattices(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(B),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(B),u1_struct_0(B)) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> k3_knaster(B,k1_abian(u1_struct_0(B),C,A),D) = k4_knaster(B,C,D,A) ) ) ) ) ).
fof(t23_knaster,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& l3_lattices(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(B),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(B),u1_struct_0(B)) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> k3_knaster(B,k1_abian(u1_struct_0(B),C,A),D) = k5_knaster(B,C,D,A) ) ) ) ) ).
fof(d8_knaster,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v2_knaster(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( r2_hidden(C,B)
& r2_hidden(D,B)
& ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ~ ( r2_hidden(E,B)
& r1_lattices(A,C,E)
& r1_lattices(A,D,E)
& ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ( ( r2_hidden(F,B)
& r1_lattices(A,C,F)
& r1_lattices(A,D,F) )
=> r1_lattices(A,E,F) ) ) ) ) ) ) ) ) ) ) ).
fof(d9_knaster,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v3_knaster(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( r2_hidden(C,B)
& r2_hidden(D,B)
& ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ~ ( r2_hidden(E,B)
& r1_lattices(A,E,C)
& r1_lattices(A,E,D)
& ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ( ( r2_hidden(F,B)
& r1_lattices(A,F,C)
& r1_lattices(A,F,D) )
=> r1_lattices(A,F,E) ) ) ) ) ) ) ) ) ) ) ).
fof(d10_knaster,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v2_knaster(B,A)
& v3_knaster(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v3_lattices(C)
& v10_lattices(C)
& l3_lattices(C) )
=> ( C = k8_knaster(A,B)
<=> ( u1_struct_0(C) = B
& ! [D] :
( m1_subset_1(D,u1_struct_0(C))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(C))
=> ? [F] :
( m1_subset_1(F,u1_struct_0(A))
& ? [G] :
( m1_subset_1(G,u1_struct_0(A))
& D = F
& E = G
& ( r3_lattices(C,D,E)
=> r3_lattices(A,F,G) )
& ( r3_lattices(A,F,G)
=> r3_lattices(C,D,E) ) ) ) ) ) ) ) ) ) ) ).
fof(t24_knaster,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& l3_lattices(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v14_quantal1(B,A)
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ? [C] :
( m1_subset_1(C,u1_struct_0(A))
& r2_abian(u1_struct_0(A),C,B) ) ) ) ).
fof(t25_knaster,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& l3_lattices(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v14_quantal1(B,A)
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r3_lattices(A,C,k3_knaster(A,B,C))
=> ! [D] :
( v3_ordinal1(D)
=> r3_lattices(A,C,k6_knaster(A,B,C,D)) ) ) ) ) ) ).
fof(t26_knaster,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& l3_lattices(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v14_quantal1(B,A)
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r3_lattices(A,k3_knaster(A,B,C),C)
=> ! [D] :
( v3_ordinal1(D)
=> r3_lattices(A,k7_knaster(A,B,C,D),C) ) ) ) ) ) ).
fof(t27_knaster,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& l3_lattices(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v14_quantal1(B,A)
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r3_lattices(A,C,k3_knaster(A,B,C))
=> ! [D] :
( v3_ordinal1(D)
=> ! [E] :
( v3_ordinal1(E)
=> ( r1_ordinal1(D,E)
=> r3_lattices(A,k6_knaster(A,B,C,D),k6_knaster(A,B,C,E)) ) ) ) ) ) ) ) ).
fof(t28_knaster,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& l3_lattices(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v14_quantal1(B,A)
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r3_lattices(A,k3_knaster(A,B,C),C)
=> ! [D] :
( v3_ordinal1(D)
=> ! [E] :
( v3_ordinal1(E)
=> ( r1_ordinal1(D,E)
=> r3_lattices(A,k7_knaster(A,B,C,E),k7_knaster(A,B,C,D)) ) ) ) ) ) ) ) ).
fof(t29_knaster,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& l3_lattices(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v14_quantal1(B,A)
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r3_lattices(A,C,k3_knaster(A,B,C))
=> ! [D] :
( v3_ordinal1(D)
=> ! [E] :
( v3_ordinal1(E)
=> ~ ( r2_xboole_0(D,E)
& ~ r2_abian(u1_struct_0(A),k6_knaster(A,B,C,E),B)
& k6_knaster(A,B,C,D) = k6_knaster(A,B,C,E) ) ) ) ) ) ) ) ).
fof(t30_knaster,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& l3_lattices(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v14_quantal1(B,A)
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r3_lattices(A,k3_knaster(A,B,C),C)
=> ! [D] :
( v3_ordinal1(D)
=> ! [E] :
( v3_ordinal1(E)
=> ~ ( r2_xboole_0(D,E)
& ~ r2_abian(u1_struct_0(A),k7_knaster(A,B,C,E),B)
& k7_knaster(A,B,C,D) = k7_knaster(A,B,C,E) ) ) ) ) ) ) ) ).
fof(t31_knaster,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& v4_lattice3(B)
& l3_lattices(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(B),u1_struct_0(B))
& v14_quantal1(C,B)
& m2_relset_1(C,u1_struct_0(B),u1_struct_0(B)) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ( ( r3_lattices(B,D,k3_knaster(B,C,D))
& r2_abian(u1_struct_0(B),k6_knaster(B,C,D,A),C) )
=> ! [E] :
( v3_ordinal1(E)
=> ( r1_ordinal1(A,E)
=> k6_knaster(B,C,D,A) = k6_knaster(B,C,D,E) ) ) ) ) ) ) ) ).
fof(t32_knaster,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& v4_lattice3(B)
& l3_lattices(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(B),u1_struct_0(B))
& v14_quantal1(C,B)
& m2_relset_1(C,u1_struct_0(B),u1_struct_0(B)) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ( ( r3_lattices(B,k3_knaster(B,C,D),D)
& r2_abian(u1_struct_0(B),k7_knaster(B,C,D,A),C) )
=> ! [E] :
( v3_ordinal1(E)
=> ( r1_ordinal1(A,E)
=> k7_knaster(B,C,D,A) = k7_knaster(B,C,D,E) ) ) ) ) ) ) ) ).
fof(t33_knaster,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& l3_lattices(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v14_quantal1(B,A)
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( r3_lattices(A,C,k3_knaster(A,B,C))
& ! [D] :
( v3_ordinal1(D)
=> ~ ( r1_tarski(k1_card_1(D),k1_card_1(u1_struct_0(A)))
& r2_abian(u1_struct_0(A),k6_knaster(A,B,C,D),B) ) ) ) ) ) ) ).
fof(t34_knaster,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& l3_lattices(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v14_quantal1(B,A)
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( r3_lattices(A,k3_knaster(A,B,C),C)
& ! [D] :
( v3_ordinal1(D)
=> ~ ( r1_tarski(k1_card_1(D),k1_card_1(u1_struct_0(A)))
& r2_abian(u1_struct_0(A),k7_knaster(A,B,C,D),B) ) ) ) ) ) ) ).
fof(t35_knaster,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& l3_lattices(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v14_quantal1(B,A)
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( r2_abian(u1_struct_0(A),C,B)
& r2_abian(u1_struct_0(A),D,B)
& ! [E] :
( v3_ordinal1(E)
=> ~ ( r1_tarski(k1_card_1(E),k1_card_1(u1_struct_0(A)))
& r2_abian(u1_struct_0(A),k6_knaster(A,B,k3_lattices(A,C,D),E),B)
& r3_lattices(A,C,k6_knaster(A,B,k3_lattices(A,C,D),E))
& r3_lattices(A,D,k6_knaster(A,B,k3_lattices(A,C,D),E)) ) ) ) ) ) ) ) ).
fof(t36_knaster,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& l3_lattices(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v14_quantal1(B,A)
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( r2_abian(u1_struct_0(A),C,B)
& r2_abian(u1_struct_0(A),D,B)
& ! [E] :
( v3_ordinal1(E)
=> ~ ( r1_tarski(k1_card_1(E),k1_card_1(u1_struct_0(A)))
& r2_abian(u1_struct_0(A),k7_knaster(A,B,k4_lattices(A,C,D),E),B)
& r3_lattices(A,k7_knaster(A,B,k4_lattices(A,C,D),E),C)
& r3_lattices(A,k7_knaster(A,B,k4_lattices(A,C,D),E),D) ) ) ) ) ) ) ) ).
fof(t37_knaster,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& l3_lattices(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v14_quantal1(B,A)
& m2_relset_1(B,u1_struct_0(A),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,C,k3_knaster(A,B,C))
& r3_lattices(A,C,D)
& r2_abian(u1_struct_0(A),D,B) )
=> ! [E] :
( v3_ordinal1(E)
=> r3_lattices(A,k6_knaster(A,B,C,E),D) ) ) ) ) ) ) ).
fof(t38_knaster,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& l3_lattices(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v14_quantal1(B,A)
& m2_relset_1(B,u1_struct_0(A),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,k3_knaster(A,B,C),C)
& r3_lattices(A,D,C)
& r2_abian(u1_struct_0(A),D,B) )
=> ! [E] :
( v3_ordinal1(E)
=> r3_lattices(A,D,k7_knaster(A,B,C,E)) ) ) ) ) ) ) ).
fof(t40_knaster,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& l3_lattices(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v14_quantal1(B,A)
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> r1_tarski(u1_struct_0(k9_knaster(A,B)),u1_struct_0(A)) ) ) ).
fof(t41_knaster,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& l3_lattices(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v14_quantal1(B,A)
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r2_hidden(C,u1_struct_0(k9_knaster(A,B)))
<=> r2_abian(u1_struct_0(A),C,B) ) ) ) ) ).
fof(t42_knaster,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& l3_lattices(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v14_quantal1(B,A)
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k9_knaster(A,B)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k9_knaster(A,B)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ( ( C = E
& D = F )
=> ( r3_lattices(k9_knaster(A,B),C,D)
<=> r3_lattices(A,E,F) ) ) ) ) ) ) ) ) ).
fof(t43_knaster,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& l3_lattices(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v14_quantal1(B,A)
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> v4_lattice3(k9_knaster(A,B)) ) ) ).
fof(d12_knaster,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& l3_lattices(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v14_quantal1(B,A)
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> k10_knaster(A,B) = k6_knaster(A,B,k5_lattices(A),k2_card_1(u1_struct_0(A))) ) ) ).
fof(d13_knaster,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& l3_lattices(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v14_quantal1(B,A)
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> k11_knaster(A,B) = k7_knaster(A,B,k6_lattices(A),k2_card_1(u1_struct_0(A))) ) ) ).
fof(t44_knaster,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& l3_lattices(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v14_quantal1(B,A)
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ( r2_abian(u1_struct_0(A),k10_knaster(A,B),B)
& ? [C] :
( v3_ordinal1(C)
& r1_tarski(k1_card_1(C),k1_card_1(u1_struct_0(A)))
& k6_knaster(A,B,k5_lattices(A),C) = k10_knaster(A,B) ) ) ) ) ).
fof(t45_knaster,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& l3_lattices(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v14_quantal1(B,A)
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ( r2_abian(u1_struct_0(A),k11_knaster(A,B),B)
& ? [C] :
( v3_ordinal1(C)
& r1_tarski(k1_card_1(C),k1_card_1(u1_struct_0(A)))
& k7_knaster(A,B,k6_lattices(A),C) = k11_knaster(A,B) ) ) ) ) ).
fof(t46_knaster,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& l3_lattices(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v14_quantal1(B,A)
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r2_abian(u1_struct_0(A),C,B)
=> ( r3_lattices(A,k10_knaster(A,B),C)
& r3_lattices(A,C,k11_knaster(A,B)) ) ) ) ) ) ).
fof(t47_knaster,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& l3_lattices(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v14_quantal1(B,A)
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r3_lattices(A,k3_knaster(A,B,C),C)
=> r3_lattices(A,k10_knaster(A,B),C) ) ) ) ) ).
fof(t48_knaster,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& l3_lattices(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v14_quantal1(B,A)
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r3_lattices(A,C,k3_knaster(A,B,C))
=> r3_lattices(A,C,k11_knaster(A,B)) ) ) ) ) ).
fof(t49_knaster,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(k1_lattice3(A)),u1_struct_0(k1_lattice3(A)))
& m2_relset_1(B,u1_struct_0(k1_lattice3(A)),u1_struct_0(k1_lattice3(A))) )
=> ( v14_quantal1(B,k1_lattice3(A))
<=> v6_cohsp_1(B) ) ) ) ).
fof(t50_knaster,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(k1_lattice3(A)),u1_struct_0(k1_lattice3(A)))
& v14_quantal1(B,k1_lattice3(A))
& m2_relset_1(B,u1_struct_0(k1_lattice3(A)),u1_struct_0(k1_lattice3(A))) )
=> ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,k1_zfmisc_1(A),k1_zfmisc_1(A))
& v6_cohsp_1(C)
& m2_relset_1(C,k1_zfmisc_1(A),k1_zfmisc_1(A))
& k1_knaster(A,C) = k10_knaster(k1_lattice3(A),B) ) ) ) ).
fof(t51_knaster,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(k1_lattice3(A)),u1_struct_0(k1_lattice3(A)))
& v14_quantal1(B,k1_lattice3(A))
& m2_relset_1(B,u1_struct_0(k1_lattice3(A)),u1_struct_0(k1_lattice3(A))) )
=> ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,k1_zfmisc_1(A),k1_zfmisc_1(A))
& v6_cohsp_1(C)
& m2_relset_1(C,k1_zfmisc_1(A),k1_zfmisc_1(A))
& k2_knaster(A,C) = k11_knaster(k1_lattice3(A),B) ) ) ) ).
fof(s1_knaster,axiom,
( ( ! [A,B] :
( r1_tarski(A,B)
=> r1_tarski(f2_s1_knaster(A),f2_s1_knaster(B)) )
& r1_tarski(f2_s1_knaster(f1_s1_knaster),f1_s1_knaster) )
=> ? [A] :
( f2_s1_knaster(A) = A
& r1_tarski(A,f1_s1_knaster) ) ) ).
fof(redefinition_v1_knaster,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& v1_funct_1(C)
& v1_funct_2(C,A,B)
& m1_relset_1(C,A,B) )
=> ( v1_knaster(C,A,B)
<=> v6_cohsp_1(C) ) ) ).
fof(dt_k1_knaster,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
& v6_cohsp_1(B)
& m1_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A)) )
=> m1_subset_1(k1_knaster(A,B),k1_zfmisc_1(A)) ) ).
fof(dt_k2_knaster,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
& v6_cohsp_1(B)
& m1_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A)) )
=> m1_subset_1(k2_knaster(A,B),k1_zfmisc_1(A)) ) ).
fof(dt_k3_knaster,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l3_lattices(A)
& v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& m1_relset_1(B,u1_struct_0(A),u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> m1_subset_1(k3_knaster(A,B,C),u1_struct_0(A)) ) ).
fof(redefinition_k3_knaster,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l3_lattices(A)
& v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& m1_relset_1(B,u1_struct_0(A),u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> k3_knaster(A,B,C) = k1_funct_1(B,C) ) ).
fof(dt_k4_knaster,axiom,
$true ).
fof(dt_k5_knaster,axiom,
$true ).
fof(dt_k6_knaster,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A)
& v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& m1_relset_1(B,u1_struct_0(A),u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A))
& v3_ordinal1(D) )
=> m1_subset_1(k6_knaster(A,B,C,D),u1_struct_0(A)) ) ).
fof(redefinition_k6_knaster,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A)
& v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& m1_relset_1(B,u1_struct_0(A),u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A))
& v3_ordinal1(D) )
=> k6_knaster(A,B,C,D) = k4_knaster(A,B,C,D) ) ).
fof(dt_k7_knaster,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A)
& v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& m1_relset_1(B,u1_struct_0(A),u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A))
& v3_ordinal1(D) )
=> m1_subset_1(k7_knaster(A,B,C,D),u1_struct_0(A)) ) ).
fof(redefinition_k7_knaster,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A)
& v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& m1_relset_1(B,u1_struct_0(A),u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A))
& v3_ordinal1(D) )
=> k7_knaster(A,B,C,D) = k5_knaster(A,B,C,D) ) ).
fof(dt_k8_knaster,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A)
& ~ v1_xboole_0(B)
& v2_knaster(B,A)
& v3_knaster(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( ~ v3_struct_0(k8_knaster(A,B))
& v3_lattices(k8_knaster(A,B))
& v10_lattices(k8_knaster(A,B))
& l3_lattices(k8_knaster(A,B)) ) ) ).
fof(dt_k9_knaster,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& l3_lattices(A)
& v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& m1_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ( ~ v3_struct_0(k9_knaster(A,B))
& v3_lattices(k9_knaster(A,B))
& v10_lattices(k9_knaster(A,B))
& l3_lattices(k9_knaster(A,B)) ) ) ).
fof(dt_k10_knaster,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& l3_lattices(A)
& v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v14_quantal1(B,A)
& m1_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> m1_subset_1(k10_knaster(A,B),u1_struct_0(A)) ) ).
fof(dt_k11_knaster,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& l3_lattices(A)
& v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v14_quantal1(B,A)
& m1_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> m1_subset_1(k11_knaster(A,B),u1_struct_0(A)) ) ).
fof(d4_knaster,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
& v6_cohsp_1(B)
& m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A)) )
=> k1_knaster(A,B) = k1_setfam_1(a_2_0_knaster(A,B)) ) ).
fof(d5_knaster,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
& v6_cohsp_1(B)
& m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A)) )
=> k2_knaster(A,B) = k3_tarski(a_2_1_knaster(A,B)) ) ).
fof(d11_knaster,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& l3_lattices(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ( v14_quantal1(B,A)
=> ! [C] :
( ( ~ v3_struct_0(C)
& v3_lattices(C)
& v10_lattices(C)
& l3_lattices(C) )
=> ( C = k9_knaster(A,B)
<=> ? [D] :
( ~ v1_xboole_0(D)
& v2_knaster(D,A)
& v3_knaster(D,A)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
& D = a_2_2_knaster(A,B)
& C = k8_knaster(A,D) ) ) ) ) ) ) ).
fof(t39_knaster,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& l3_lattices(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v14_quantal1(B,A)
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> u1_struct_0(k9_knaster(A,B)) = a_2_3_knaster(A,B) ) ) ).
fof(fraenkel_a_2_0_knaster,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k1_zfmisc_1(B),k1_zfmisc_1(B))
& v6_cohsp_1(C)
& m2_relset_1(C,k1_zfmisc_1(B),k1_zfmisc_1(B)) )
=> ( r2_hidden(A,a_2_0_knaster(B,C))
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(B))
& A = D
& r1_tarski(k8_funct_2(k1_zfmisc_1(B),k1_zfmisc_1(B),C,D),D) ) ) ) ).
fof(fraenkel_a_2_1_knaster,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k1_zfmisc_1(B),k1_zfmisc_1(B))
& v6_cohsp_1(C)
& m2_relset_1(C,k1_zfmisc_1(B),k1_zfmisc_1(B)) )
=> ( r2_hidden(A,a_2_1_knaster(B,C))
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(B))
& A = D
& r1_tarski(D,k8_funct_2(k1_zfmisc_1(B),k1_zfmisc_1(B),C,D)) ) ) ) ).
fof(fraenkel_a_2_2_knaster,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& v4_lattice3(B)
& l3_lattices(B)
& v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(B),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(B),u1_struct_0(B)) )
=> ( r2_hidden(A,a_2_2_knaster(B,C))
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(B))
& A = D
& r2_abian(u1_struct_0(B),D,C) ) ) ) ).
fof(fraenkel_a_2_3_knaster,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& v4_lattice3(B)
& l3_lattices(B)
& v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(B),u1_struct_0(B))
& v14_quantal1(C,B)
& m2_relset_1(C,u1_struct_0(B),u1_struct_0(B)) )
=> ( r2_hidden(A,a_2_3_knaster(B,C))
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(B))
& A = D
& r2_abian(u1_struct_0(B),D,C) ) ) ) ).
%------------------------------------------------------------------------------