SET007 Axioms: SET007+270.ax
%------------------------------------------------------------------------------
% File : SET007+270 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Algebra of Normal Forms
% 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 : normform [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 129 ( 38 unt; 0 def)
% Number of atoms : 599 ( 64 equ)
% Maximal formula atoms : 12 ( 4 avg)
% Number of connectives : 561 ( 91 ~; 0 |; 210 &)
% ( 10 <=>; 250 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 7 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 33 ( 31 usr; 1 prp; 0-4 aty)
% Number of functors : 41 ( 41 usr; 1 con; 0-6 aty)
% Number of variables : 342 ( 337 !; 5 ?)
% 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_normform,axiom,
! [A] :
( v1_relat_1(k5_normform(A))
& v1_funct_1(k5_normform(A))
& v1_funct_2(k5_normform(A),k2_zfmisc_1(k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A))),k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)))
& v1_binop_1(k5_normform(A),k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)))
& v2_binop_1(k5_normform(A),k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)))
& v3_binop_1(k5_normform(A),k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A))) ) ).
fof(fc2_normform,axiom,
! [A] :
( v1_relat_1(k7_normform(A))
& ~ v1_xboole_0(k7_normform(A)) ) ).
fof(fc3_normform,axiom,
! [A] : ~ v1_xboole_0(k8_normform(A)) ).
fof(fc4_normform,axiom,
! [A] :
( ~ v3_struct_0(k12_normform(A))
& v3_lattices(k12_normform(A)) ) ).
fof(fc5_normform,axiom,
! [A] :
( ~ v3_struct_0(k12_normform(A))
& v3_lattices(k12_normform(A))
& v4_lattices(k12_normform(A))
& v5_lattices(k12_normform(A))
& v6_lattices(k12_normform(A))
& v7_lattices(k12_normform(A))
& v8_lattices(k12_normform(A))
& v9_lattices(k12_normform(A))
& v10_lattices(k12_normform(A)) ) ).
fof(fc6_normform,axiom,
! [A] :
( ~ v3_struct_0(k12_normform(A))
& v3_lattices(k12_normform(A))
& v4_lattices(k12_normform(A))
& v5_lattices(k12_normform(A))
& v6_lattices(k12_normform(A))
& v7_lattices(k12_normform(A))
& v8_lattices(k12_normform(A))
& v9_lattices(k12_normform(A))
& v10_lattices(k12_normform(A))
& v11_lattices(k12_normform(A))
& v12_lattices(k12_normform(A))
& v13_lattices(k12_normform(A)) ) ).
fof(d1_normform,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_finsub_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v4_finsub_1(B) )
=> ! [C] :
( m1_subset_1(C,k2_zfmisc_1(A,B))
=> ! [D] :
( m1_subset_1(D,k2_zfmisc_1(A,B))
=> ( r1_normform(A,B,C,D)
<=> ( r1_tarski(k2_domain_1(A,B,C),k2_domain_1(A,B,D))
& r1_tarski(k3_domain_1(A,B,C),k3_domain_1(A,B,D)) ) ) ) ) ) ) ).
fof(d2_normform,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_finsub_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v4_finsub_1(B) )
=> ! [C] :
( m1_subset_1(C,k2_zfmisc_1(A,B))
=> ! [D] :
( m1_subset_1(D,k2_zfmisc_1(A,B))
=> k1_normform(A,B,C,D) = k1_domain_1(A,B,k1_finsub_1(A,k2_domain_1(A,B,C),k2_domain_1(A,B,D)),k1_finsub_1(B,k3_domain_1(A,B,C),k3_domain_1(A,B,D))) ) ) ) ) ).
fof(d3_normform,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_finsub_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v4_finsub_1(B) )
=> ! [C] :
( m1_subset_1(C,k2_zfmisc_1(A,B))
=> ! [D] :
( m1_subset_1(D,k2_zfmisc_1(A,B))
=> k2_normform(A,B,C,D) = k1_domain_1(A,B,k3_finsub_1(A,k2_domain_1(A,B,C),k2_domain_1(A,B,D)),k3_finsub_1(B,k3_domain_1(A,B,C),k3_domain_1(A,B,D))) ) ) ) ) ).
fof(d4_normform,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_finsub_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v4_finsub_1(B) )
=> ! [C] :
( m1_subset_1(C,k2_zfmisc_1(A,B))
=> ! [D] :
( m1_subset_1(D,k2_zfmisc_1(A,B))
=> k3_normform(A,B,C,D) = k1_domain_1(A,B,k2_finsub_1(A,k2_domain_1(A,B,C),k2_domain_1(A,B,D)),k2_finsub_1(B,k3_domain_1(A,B,C),k3_domain_1(A,B,D))) ) ) ) ) ).
fof(d5_normform,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_finsub_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v4_finsub_1(B) )
=> ! [C] :
( m1_subset_1(C,k2_zfmisc_1(A,B))
=> ! [D] :
( m1_subset_1(D,k2_zfmisc_1(A,B))
=> k4_normform(A,B,C,D) = k1_domain_1(A,B,k4_finsub_1(A,k2_domain_1(A,B,C),k2_domain_1(A,B,D)),k4_finsub_1(B,k3_domain_1(A,B,C),k3_domain_1(A,B,D))) ) ) ) ) ).
fof(t1_normform,axiom,
$true ).
fof(t2_normform,axiom,
$true ).
fof(t3_normform,axiom,
$true ).
fof(t4_normform,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_finsub_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v4_finsub_1(B) )
=> ! [C] :
( m1_subset_1(C,k2_zfmisc_1(A,B))
=> ! [D] :
( m1_subset_1(D,k2_zfmisc_1(A,B))
=> ( ( r1_normform(A,B,C,D)
& r1_normform(A,B,D,C) )
=> C = D ) ) ) ) ) ).
fof(t5_normform,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_finsub_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v4_finsub_1(B) )
=> ! [C] :
( m1_subset_1(C,k2_zfmisc_1(A,B))
=> ! [D] :
( m1_subset_1(D,k2_zfmisc_1(A,B))
=> ! [E] :
( m1_subset_1(E,k2_zfmisc_1(A,B))
=> ( ( r1_normform(A,B,C,D)
& r1_normform(A,B,D,E) )
=> r1_normform(A,B,C,E) ) ) ) ) ) ) ).
fof(t6_normform,axiom,
$true ).
fof(t7_normform,axiom,
$true ).
fof(t8_normform,axiom,
$true ).
fof(t9_normform,axiom,
$true ).
fof(t10_normform,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_finsub_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v4_finsub_1(B) )
=> ! [C] :
( m1_subset_1(C,k2_zfmisc_1(A,B))
=> ! [D] :
( m1_subset_1(D,k2_zfmisc_1(A,B))
=> ( k2_domain_1(A,B,k1_normform(A,B,C,D)) = k1_finsub_1(A,k2_domain_1(A,B,C),k2_domain_1(A,B,D))
& k3_domain_1(A,B,k1_normform(A,B,C,D)) = k1_finsub_1(B,k3_domain_1(A,B,C),k3_domain_1(A,B,D)) ) ) ) ) ) ).
fof(t11_normform,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_finsub_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v4_finsub_1(B) )
=> ! [C] :
( m1_subset_1(C,k2_zfmisc_1(A,B))
=> ! [D] :
( m1_subset_1(D,k2_zfmisc_1(A,B))
=> ( k2_domain_1(A,B,k2_normform(A,B,C,D)) = k3_finsub_1(A,k2_domain_1(A,B,C),k2_domain_1(A,B,D))
& k3_domain_1(A,B,k2_normform(A,B,C,D)) = k3_finsub_1(B,k3_domain_1(A,B,C),k3_domain_1(A,B,D)) ) ) ) ) ) ).
fof(t12_normform,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_finsub_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v4_finsub_1(B) )
=> ! [C] :
( m1_subset_1(C,k2_zfmisc_1(A,B))
=> ! [D] :
( m1_subset_1(D,k2_zfmisc_1(A,B))
=> ( k2_domain_1(A,B,k3_normform(A,B,C,D)) = k2_finsub_1(A,k2_domain_1(A,B,C),k2_domain_1(A,B,D))
& k3_domain_1(A,B,k3_normform(A,B,C,D)) = k2_finsub_1(B,k3_domain_1(A,B,C),k3_domain_1(A,B,D)) ) ) ) ) ) ).
fof(t13_normform,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_finsub_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v4_finsub_1(B) )
=> ! [C] :
( m1_subset_1(C,k2_zfmisc_1(A,B))
=> ! [D] :
( m1_subset_1(D,k2_zfmisc_1(A,B))
=> ( k2_domain_1(A,B,k4_normform(A,B,C,D)) = k4_finsub_1(A,k2_domain_1(A,B,C),k2_domain_1(A,B,D))
& k3_domain_1(A,B,k4_normform(A,B,C,D)) = k4_finsub_1(B,k3_domain_1(A,B,C),k3_domain_1(A,B,D)) ) ) ) ) ) ).
fof(t14_normform,axiom,
$true ).
fof(t15_normform,axiom,
$true ).
fof(t16_normform,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_finsub_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v4_finsub_1(B) )
=> ! [C] :
( m1_subset_1(C,k2_zfmisc_1(A,B))
=> ! [D] :
( m1_subset_1(D,k2_zfmisc_1(A,B))
=> ! [E] :
( m1_subset_1(E,k2_zfmisc_1(A,B))
=> k1_normform(A,B,k1_normform(A,B,C,D),E) = k1_normform(A,B,C,k1_normform(A,B,D,E)) ) ) ) ) ) ).
fof(t17_normform,axiom,
$true ).
fof(t18_normform,axiom,
$true ).
fof(t19_normform,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_finsub_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v4_finsub_1(B) )
=> ! [C] :
( m1_subset_1(C,k2_zfmisc_1(A,B))
=> ! [D] :
( m1_subset_1(D,k2_zfmisc_1(A,B))
=> ! [E] :
( m1_subset_1(E,k2_zfmisc_1(A,B))
=> k2_normform(A,B,k2_normform(A,B,C,D),E) = k2_normform(A,B,C,k2_normform(A,B,D,E)) ) ) ) ) ) ).
fof(t20_normform,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_finsub_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v4_finsub_1(B) )
=> ! [C] :
( m1_subset_1(C,k2_zfmisc_1(A,B))
=> ! [D] :
( m1_subset_1(D,k2_zfmisc_1(A,B))
=> ! [E] :
( m1_subset_1(E,k2_zfmisc_1(A,B))
=> k2_normform(A,B,C,k1_normform(A,B,D,E)) = k1_normform(A,B,k2_normform(A,B,C,D),k2_normform(A,B,C,E)) ) ) ) ) ) ).
fof(t21_normform,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_finsub_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v4_finsub_1(B) )
=> ! [C] :
( m1_subset_1(C,k2_zfmisc_1(A,B))
=> ! [D] :
( m1_subset_1(D,k2_zfmisc_1(A,B))
=> k1_normform(A,B,C,k2_normform(A,B,D,C)) = C ) ) ) ) ).
fof(t22_normform,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_finsub_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v4_finsub_1(B) )
=> ! [C] :
( m1_subset_1(C,k2_zfmisc_1(A,B))
=> ! [D] :
( m1_subset_1(D,k2_zfmisc_1(A,B))
=> k2_normform(A,B,C,k1_normform(A,B,D,C)) = C ) ) ) ) ).
fof(t23_normform,axiom,
$true ).
fof(t24_normform,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_finsub_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v4_finsub_1(B) )
=> ! [C] :
( m1_subset_1(C,k2_zfmisc_1(A,B))
=> ! [D] :
( m1_subset_1(D,k2_zfmisc_1(A,B))
=> ! [E] :
( m1_subset_1(E,k2_zfmisc_1(A,B))
=> k1_normform(A,B,C,k2_normform(A,B,D,E)) = k2_normform(A,B,k1_normform(A,B,C,D),k1_normform(A,B,C,E)) ) ) ) ) ) ).
fof(t25_normform,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_finsub_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v4_finsub_1(B) )
=> ! [C] :
( m1_subset_1(C,k2_zfmisc_1(A,B))
=> ! [D] :
( m1_subset_1(D,k2_zfmisc_1(A,B))
=> ! [E] :
( m1_subset_1(E,k2_zfmisc_1(A,B))
=> ( ( r1_normform(A,B,C,D)
& r1_normform(A,B,E,D) )
=> r1_normform(A,B,k1_normform(A,B,C,E),D) ) ) ) ) ) ) ).
fof(t26_normform,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_finsub_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v4_finsub_1(B) )
=> ! [C] :
( m1_subset_1(C,k2_zfmisc_1(A,B))
=> ! [D] :
( m1_subset_1(D,k2_zfmisc_1(A,B))
=> ( r1_normform(A,B,C,k1_normform(A,B,C,D))
& r1_normform(A,B,D,k1_normform(A,B,C,D)) ) ) ) ) ) ).
fof(t27_normform,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_finsub_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v4_finsub_1(B) )
=> ! [C] :
( m1_subset_1(C,k2_zfmisc_1(A,B))
=> ! [D] :
( m1_subset_1(D,k2_zfmisc_1(A,B))
=> ( C = k1_normform(A,B,C,D)
=> r1_normform(A,B,D,C) ) ) ) ) ) ).
fof(t28_normform,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_finsub_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v4_finsub_1(B) )
=> ! [C] :
( m1_subset_1(C,k2_zfmisc_1(A,B))
=> ! [D] :
( m1_subset_1(D,k2_zfmisc_1(A,B))
=> ! [E] :
( m1_subset_1(E,k2_zfmisc_1(A,B))
=> ( r1_normform(A,B,C,D)
=> ( r1_normform(A,B,k1_normform(A,B,E,C),k1_normform(A,B,E,D))
& r1_normform(A,B,k1_normform(A,B,C,E),k1_normform(A,B,D,E)) ) ) ) ) ) ) ) ).
fof(t29_normform,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_finsub_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v4_finsub_1(B) )
=> ! [C] :
( m1_subset_1(C,k2_zfmisc_1(A,B))
=> ! [D] :
( m1_subset_1(D,k2_zfmisc_1(A,B))
=> k1_normform(A,B,k3_normform(A,B,C,D),D) = k1_normform(A,B,C,D) ) ) ) ) ).
fof(t30_normform,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_finsub_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v4_finsub_1(B) )
=> ! [C] :
( m1_subset_1(C,k2_zfmisc_1(A,B))
=> ! [D] :
( m1_subset_1(D,k2_zfmisc_1(A,B))
=> ! [E] :
( m1_subset_1(E,k2_zfmisc_1(A,B))
=> ( r1_normform(A,B,k3_normform(A,B,C,D),E)
=> r1_normform(A,B,C,k1_normform(A,B,D,E)) ) ) ) ) ) ) ).
fof(t31_normform,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_finsub_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v4_finsub_1(B) )
=> ! [C] :
( m1_subset_1(C,k2_zfmisc_1(A,B))
=> ! [D] :
( m1_subset_1(D,k2_zfmisc_1(A,B))
=> ! [E] :
( m1_subset_1(E,k2_zfmisc_1(A,B))
=> ( r1_normform(A,B,C,k1_normform(A,B,D,E))
=> r1_normform(A,B,k3_normform(A,B,C,E),D) ) ) ) ) ) ) ).
fof(d6_normform,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A))),k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)))
& m2_relset_1(B,k2_zfmisc_1(k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A))),k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A))) )
=> ( B = k5_normform(A)
<=> ! [C] :
( m1_subset_1(C,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)))
=> ! [D] :
( m1_subset_1(D,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)))
=> k2_binop_1(k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),B,C,D) = k1_normform(k5_finsub_1(A),k5_finsub_1(A),C,D) ) ) ) ) ).
fof(d7_normform,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B,C] :
( m1_subset_1(C,k5_finsub_1(A))
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,k2_zfmisc_1(k5_finsub_1(B),k5_finsub_1(B)))
& m2_relset_1(D,A,k2_zfmisc_1(k5_finsub_1(B),k5_finsub_1(B))) )
=> k6_normform(A,B,C,D) = k7_setwiseo(A,k2_zfmisc_1(k5_finsub_1(B),k5_finsub_1(B)),k5_normform(B),C,D) ) ) ) ).
fof(t32_normform,axiom,
$true ).
fof(t33_normform,axiom,
$true ).
fof(t34_normform,axiom,
$true ).
fof(t35_normform,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,B,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)))
& m2_relset_1(C,B,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A))) )
=> ! [D] :
( m1_subset_1(D,k5_finsub_1(B))
=> ! [E] :
( m1_subset_1(E,B)
=> ( r2_hidden(E,D)
=> r1_normform(k5_finsub_1(A),k5_finsub_1(A),k8_funct_2(B,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),C,E),k6_normform(B,A,D,C)) ) ) ) ) ) ).
fof(t36_normform,axiom,
! [A] : r3_binop_1(k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k1_domain_1(k5_finsub_1(A),k5_finsub_1(A),k1_setwiseo(A),k1_setwiseo(A)),k5_normform(A)) ).
fof(t37_normform,axiom,
! [A] : v1_setwiseo(k5_normform(A),k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A))) ).
fof(t38_normform,axiom,
! [A] : k3_binop_1(k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k5_normform(A)) = k1_domain_1(k5_finsub_1(A),k5_finsub_1(A),k1_setwiseo(A),k1_setwiseo(A)) ).
fof(t39_normform,axiom,
! [A,B] :
( m1_subset_1(B,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)))
=> r1_normform(k5_finsub_1(A),k5_finsub_1(A),k3_binop_1(k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k5_normform(A)),B) ) ).
fof(t40_normform,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,B,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)))
& m2_relset_1(C,B,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A))) )
=> ! [D] :
( m1_subset_1(D,k5_finsub_1(B))
=> ! [E] :
( m1_subset_1(E,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)))
=> ( ! [F] :
( m1_subset_1(F,B)
=> ( r2_hidden(F,D)
=> r1_normform(k5_finsub_1(A),k5_finsub_1(A),k8_funct_2(B,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),C,F),E) ) )
=> r1_normform(k5_finsub_1(A),k5_finsub_1(A),k6_normform(B,A,D,C),E) ) ) ) ) ) ).
fof(t41_normform,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,k5_finsub_1(B))
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,B,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)))
& m2_relset_1(D,B,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A))) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,B,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)))
& m2_relset_1(E,B,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A))) )
=> ( k7_relat_1(D,C) = k7_relat_1(E,C)
=> k6_normform(B,A,C,D) = k6_normform(B,A,C,E) ) ) ) ) ) ).
fof(t42_normform,axiom,
! [A,B] :
( m1_subset_1(B,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)))
=> ( r2_hidden(B,k7_normform(A))
<=> r1_xboole_0(k2_domain_1(k5_finsub_1(A),k5_finsub_1(A),B),k3_domain_1(k5_finsub_1(A),k5_finsub_1(A),B)) ) ) ).
fof(t43_normform,axiom,
! [A,B] :
( m1_subset_1(B,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)))
=> ! [C] :
( m1_subset_1(C,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)))
=> ( ( r2_hidden(B,k7_normform(A))
& r2_hidden(C,k7_normform(A)) )
=> ( r2_hidden(k1_normform(k5_finsub_1(A),k5_finsub_1(A),B,C),k7_normform(A))
<=> k5_setwiseo(A,k3_finsub_1(k5_finsub_1(A),k2_domain_1(k5_finsub_1(A),k5_finsub_1(A),B),k3_domain_1(k5_finsub_1(A),k5_finsub_1(A),C)),k3_finsub_1(k5_finsub_1(A),k2_domain_1(k5_finsub_1(A),k5_finsub_1(A),C),k3_domain_1(k5_finsub_1(A),k5_finsub_1(A),B))) = k1_xboole_0 ) ) ) ) ).
fof(t44_normform,axiom,
! [A,B] :
( m2_subset_1(B,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k7_normform(A))
=> r1_xboole_0(k2_domain_1(k5_finsub_1(A),k5_finsub_1(A),B),k3_domain_1(k5_finsub_1(A),k5_finsub_1(A),B)) ) ).
fof(t45_normform,axiom,
! [A,B] :
( m1_subset_1(B,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)))
=> ! [C] :
( m2_subset_1(C,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k7_normform(A))
=> ( r1_normform(k5_finsub_1(A),k5_finsub_1(A),B,C)
=> m2_subset_1(B,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k7_normform(A)) ) ) ) ).
fof(t46_normform,axiom,
! [A,B] :
( m2_subset_1(B,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k7_normform(A))
=> ! [C] :
~ ( r2_hidden(C,k2_domain_1(k5_finsub_1(A),k5_finsub_1(A),B))
& r2_hidden(C,k3_domain_1(k5_finsub_1(A),k5_finsub_1(A),B)) ) ) ).
fof(t47_normform,axiom,
! [A,B] :
( m2_subset_1(B,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k7_normform(A))
=> ! [C] :
( m2_subset_1(C,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k7_normform(A))
=> ~ ( ~ r2_hidden(k1_normform(k5_finsub_1(A),k5_finsub_1(A),B,C),k7_normform(A))
& ! [D] :
( m1_subset_1(D,A)
=> ( ~ ( r2_hidden(D,k2_domain_1(k5_finsub_1(A),k5_finsub_1(A),B))
& r2_hidden(D,k3_domain_1(k5_finsub_1(A),k5_finsub_1(A),C)) )
& ~ ( r2_hidden(D,k2_domain_1(k5_finsub_1(A),k5_finsub_1(A),C))
& r2_hidden(D,k3_domain_1(k5_finsub_1(A),k5_finsub_1(A),B)) ) ) ) ) ) ) ).
fof(t48_normform,axiom,
$true ).
fof(t49_normform,axiom,
! [A,B] :
( m1_subset_1(B,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)))
=> ( r1_xboole_0(k2_domain_1(k5_finsub_1(A),k5_finsub_1(A),B),k3_domain_1(k5_finsub_1(A),k5_finsub_1(A),B))
=> m2_subset_1(B,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k7_normform(A)) ) ) ).
fof(t50_normform,axiom,
! [A,B] :
( m2_subset_1(B,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k7_normform(A))
=> ! [C,D] :
( ( r1_tarski(C,k2_domain_1(k5_finsub_1(A),k5_finsub_1(A),B))
& r1_tarski(D,k3_domain_1(k5_finsub_1(A),k5_finsub_1(A),B)) )
=> m2_subset_1(k4_tarski(C,D),k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k7_normform(A)) ) ) ).
fof(t51_normform,axiom,
! [A] : r2_hidden(k1_xboole_0,k8_normform(A)) ).
fof(t52_normform,axiom,
! [A,B] :
( m2_subset_1(B,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k7_normform(A))
=> ! [C] :
( m2_subset_1(C,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k7_normform(A))
=> ! [D] :
( m1_subset_1(D,k5_finsub_1(k7_normform(A)))
=> ( ( r2_hidden(D,k8_normform(A))
& r2_hidden(B,D)
& r2_hidden(C,D)
& r1_normform(k5_finsub_1(A),k5_finsub_1(A),B,C) )
=> B = C ) ) ) ) ).
fof(t53_normform,axiom,
! [A,B] :
( m1_subset_1(B,k5_finsub_1(k7_normform(A)))
=> ( ! [C] :
( m2_subset_1(C,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k7_normform(A))
=> ! [D] :
( m2_subset_1(D,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k7_normform(A))
=> ( ( r2_hidden(C,B)
& r2_hidden(D,B)
& r1_normform(k5_finsub_1(A),k5_finsub_1(A),C,D) )
=> C = D ) ) )
=> r2_hidden(B,k8_normform(A)) ) ) ).
fof(t54_normform,axiom,
$true ).
fof(t55_normform,axiom,
! [A,B] :
( m1_subset_1(B,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)))
=> ! [C] :
( m1_subset_1(C,k5_finsub_1(k7_normform(A)))
=> ! [D] :
( m1_subset_1(D,k5_finsub_1(k7_normform(A)))
=> ~ ( r2_hidden(B,k10_normform(A,C,D))
& ! [E] :
( m2_subset_1(E,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k7_normform(A))
=> ! [F] :
( m2_subset_1(F,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k7_normform(A))
=> ~ ( r2_hidden(E,C)
& r2_hidden(F,D)
& B = k1_normform(k5_finsub_1(A),k5_finsub_1(A),E,F) ) ) ) ) ) ) ) ).
fof(t56_normform,axiom,
! [A,B] :
( m2_subset_1(B,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k7_normform(A))
=> ! [C] :
( m2_subset_1(C,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k7_normform(A))
=> ! [D] :
( m1_subset_1(D,k5_finsub_1(k7_normform(A)))
=> ! [E] :
( m1_subset_1(E,k5_finsub_1(k7_normform(A)))
=> ( ( r2_hidden(B,D)
& r2_hidden(C,E)
& r2_hidden(k1_normform(k5_finsub_1(A),k5_finsub_1(A),B,C),k7_normform(A)) )
=> r2_hidden(k1_normform(k5_finsub_1(A),k5_finsub_1(A),B,C),k10_normform(A,D,E)) ) ) ) ) ) ).
fof(t57_normform,axiom,
$true ).
fof(t58_normform,axiom,
! [A,B] :
( m2_subset_1(B,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k7_normform(A))
=> ! [C] :
( m2_subset_1(C,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k7_normform(A))
=> ! [D] :
( m1_subset_1(D,k5_finsub_1(k7_normform(A)))
=> ( r2_hidden(B,k9_normform(A,D))
=> ( r2_hidden(B,D)
& ( ( r2_hidden(C,D)
& r1_normform(k5_finsub_1(A),k5_finsub_1(A),C,B) )
=> C = B ) ) ) ) ) ) ).
fof(t59_normform,axiom,
! [A,B] :
( m2_subset_1(B,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k7_normform(A))
=> ! [C] :
( m1_subset_1(C,k5_finsub_1(k7_normform(A)))
=> ( r2_hidden(B,k9_normform(A,C))
=> r2_hidden(B,C) ) ) ) ).
fof(t60_normform,axiom,
! [A,B] :
( m2_subset_1(B,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k7_normform(A))
=> ! [C] :
( m2_subset_1(C,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k7_normform(A))
=> ! [D] :
( m1_subset_1(D,k5_finsub_1(k7_normform(A)))
=> ( ( r2_hidden(B,k9_normform(A,D))
& r2_hidden(C,D)
& r1_normform(k5_finsub_1(A),k5_finsub_1(A),C,B) )
=> C = B ) ) ) ) ).
fof(t61_normform,axiom,
! [A,B] :
( m2_subset_1(B,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k7_normform(A))
=> ! [C] :
( m1_subset_1(C,k5_finsub_1(k7_normform(A)))
=> ( ( r2_hidden(B,C)
& ! [D] :
( m2_subset_1(D,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k7_normform(A))
=> ( ( r2_hidden(D,C)
& r1_normform(k5_finsub_1(A),k5_finsub_1(A),D,B) )
=> D = B ) ) )
=> r2_hidden(B,k9_normform(A,C)) ) ) ) ).
fof(t62_normform,axiom,
$true ).
fof(t63_normform,axiom,
$true ).
fof(t64_normform,axiom,
! [A,B] :
( m1_subset_1(B,k5_finsub_1(k7_normform(A)))
=> r1_tarski(k9_normform(A,B),B) ) ).
fof(t65_normform,axiom,
! [A,B] :
( m2_subset_1(B,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k7_normform(A))
=> ! [C] :
( m1_subset_1(C,k5_finsub_1(k7_normform(A)))
=> ~ ( r2_hidden(B,C)
& ! [D] :
( m2_subset_1(D,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k7_normform(A))
=> ~ ( r1_normform(k5_finsub_1(A),k5_finsub_1(A),D,B)
& r2_hidden(D,k9_normform(A,C)) ) ) ) ) ) ).
fof(t66_normform,axiom,
! [A,B] :
( m2_subset_1(B,k5_finsub_1(k7_normform(A)),k8_normform(A))
=> k9_normform(A,B) = B ) ).
fof(t67_normform,axiom,
! [A,B] :
( m1_subset_1(B,k5_finsub_1(k7_normform(A)))
=> ! [C] :
( m1_subset_1(C,k5_finsub_1(k7_normform(A)))
=> r1_tarski(k9_normform(A,k5_setwiseo(k7_normform(A),B,C)),k5_setwiseo(k7_normform(A),k9_normform(A,B),C)) ) ) ).
fof(t68_normform,axiom,
! [A,B] :
( m1_subset_1(B,k5_finsub_1(k7_normform(A)))
=> ! [C] :
( m1_subset_1(C,k5_finsub_1(k7_normform(A)))
=> k9_normform(A,k5_setwiseo(k7_normform(A),k9_normform(A,B),C)) = k9_normform(A,k5_setwiseo(k7_normform(A),B,C)) ) ) ).
fof(t69_normform,axiom,
! [A,B] :
( m1_subset_1(B,k5_finsub_1(k7_normform(A)))
=> ! [C] :
( m1_subset_1(C,k5_finsub_1(k7_normform(A)))
=> k9_normform(A,k5_setwiseo(k7_normform(A),B,k9_normform(A,C))) = k9_normform(A,k5_setwiseo(k7_normform(A),B,C)) ) ) ).
fof(t70_normform,axiom,
! [A,B] :
( m1_subset_1(B,k5_finsub_1(k7_normform(A)))
=> ! [C] :
( m1_subset_1(C,k5_finsub_1(k7_normform(A)))
=> ! [D] :
( m1_subset_1(D,k5_finsub_1(k7_normform(A)))
=> ( r1_tarski(B,C)
=> r1_tarski(k10_normform(A,B,D),k10_normform(A,C,D)) ) ) ) ) ).
fof(t71_normform,axiom,
! [A,B] :
( m1_subset_1(B,k5_finsub_1(k7_normform(A)))
=> ! [C] :
( m1_subset_1(C,k5_finsub_1(k7_normform(A)))
=> r1_tarski(k9_normform(A,k10_normform(A,B,C)),k10_normform(A,k9_normform(A,B),C)) ) ) ).
fof(t72_normform,axiom,
! [A,B] :
( m1_subset_1(B,k5_finsub_1(k7_normform(A)))
=> ! [C] :
( m1_subset_1(C,k5_finsub_1(k7_normform(A)))
=> k10_normform(A,B,C) = k10_normform(A,C,B) ) ) ).
fof(t73_normform,axiom,
! [A,B] :
( m1_subset_1(B,k5_finsub_1(k7_normform(A)))
=> ! [C] :
( m1_subset_1(C,k5_finsub_1(k7_normform(A)))
=> ! [D] :
( m1_subset_1(D,k5_finsub_1(k7_normform(A)))
=> ( r1_tarski(B,C)
=> r1_tarski(k10_normform(A,D,B),k10_normform(A,D,C)) ) ) ) ) ).
fof(t74_normform,axiom,
! [A,B] :
( m1_subset_1(B,k5_finsub_1(k7_normform(A)))
=> ! [C] :
( m1_subset_1(C,k5_finsub_1(k7_normform(A)))
=> k9_normform(A,k10_normform(A,k9_normform(A,B),C)) = k9_normform(A,k10_normform(A,B,C)) ) ) ).
fof(t75_normform,axiom,
! [A,B] :
( m1_subset_1(B,k5_finsub_1(k7_normform(A)))
=> ! [C] :
( m1_subset_1(C,k5_finsub_1(k7_normform(A)))
=> k9_normform(A,k10_normform(A,B,k9_normform(A,C))) = k9_normform(A,k10_normform(A,B,C)) ) ) ).
fof(t76_normform,axiom,
! [A,B] :
( m2_subset_1(B,k5_finsub_1(k7_normform(A)),k8_normform(A))
=> ! [C] :
( m2_subset_1(C,k5_finsub_1(k7_normform(A)),k8_normform(A))
=> ! [D] :
( m2_subset_1(D,k5_finsub_1(k7_normform(A)),k8_normform(A))
=> k10_normform(A,B,k10_normform(A,C,D)) = k10_normform(A,k10_normform(A,B,C),D) ) ) ) ).
fof(t77_normform,axiom,
! [A,B] :
( m2_subset_1(B,k5_finsub_1(k7_normform(A)),k8_normform(A))
=> ! [C] :
( m2_subset_1(C,k5_finsub_1(k7_normform(A)),k8_normform(A))
=> ! [D] :
( m2_subset_1(D,k5_finsub_1(k7_normform(A)),k8_normform(A))
=> k10_normform(A,B,k5_setwiseo(k7_normform(A),C,D)) = k5_setwiseo(k7_normform(A),k10_normform(A,B,C),k10_normform(A,B,D)) ) ) ) ).
fof(t78_normform,axiom,
! [A,B] :
( m1_subset_1(B,k5_finsub_1(k7_normform(A)))
=> r1_tarski(B,k10_normform(A,B,B)) ) ).
fof(t79_normform,axiom,
! [A,B] :
( m2_subset_1(B,k5_finsub_1(k7_normform(A)),k8_normform(A))
=> k9_normform(A,k10_normform(A,B,B)) = k9_normform(A,B) ) ).
fof(d12_normform,axiom,
$true ).
fof(d13_normform,axiom,
$true ).
fof(d14_normform,axiom,
! [A,B] :
( ( v3_lattices(B)
& l3_lattices(B) )
=> ( B = k12_normform(A)
<=> ( u1_struct_0(B) = k8_normform(A)
& ! [C] :
( m2_subset_1(C,k5_finsub_1(k7_normform(A)),k8_normform(A))
=> ! [D] :
( m2_subset_1(D,k5_finsub_1(k7_normform(A)),k8_normform(A))
=> ( k1_binop_1(u2_lattices(B),C,D) = k9_normform(A,k5_setwiseo(k7_normform(A),C,D))
& k1_binop_1(u1_lattices(B),C,D) = k9_normform(A,k10_normform(A,C,D)) ) ) ) ) ) ) ).
fof(t80_normform,axiom,
$true ).
fof(t81_normform,axiom,
$true ).
fof(t82_normform,axiom,
$true ).
fof(t83_normform,axiom,
$true ).
fof(t84_normform,axiom,
$true ).
fof(t85_normform,axiom,
! [A] : m1_subset_1(k1_xboole_0,u1_struct_0(k12_normform(A))) ).
fof(t86_normform,axiom,
! [A] : k5_lattices(k12_normform(A)) = k1_xboole_0 ).
fof(reflexivity_r1_normform,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& v4_finsub_1(A)
& ~ v1_xboole_0(B)
& v4_finsub_1(B)
& m1_subset_1(C,k2_zfmisc_1(A,B))
& m1_subset_1(D,k2_zfmisc_1(A,B)) )
=> r1_normform(A,B,C,C) ) ).
fof(dt_k1_normform,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& v4_finsub_1(A)
& ~ v1_xboole_0(B)
& v4_finsub_1(B)
& m1_subset_1(C,k2_zfmisc_1(A,B))
& m1_subset_1(D,k2_zfmisc_1(A,B)) )
=> m1_subset_1(k1_normform(A,B,C,D),k2_zfmisc_1(A,B)) ) ).
fof(commutativity_k1_normform,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& v4_finsub_1(A)
& ~ v1_xboole_0(B)
& v4_finsub_1(B)
& m1_subset_1(C,k2_zfmisc_1(A,B))
& m1_subset_1(D,k2_zfmisc_1(A,B)) )
=> k1_normform(A,B,C,D) = k1_normform(A,B,D,C) ) ).
fof(idempotence_k1_normform,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& v4_finsub_1(A)
& ~ v1_xboole_0(B)
& v4_finsub_1(B)
& m1_subset_1(C,k2_zfmisc_1(A,B))
& m1_subset_1(D,k2_zfmisc_1(A,B)) )
=> k1_normform(A,B,C,C) = C ) ).
fof(dt_k2_normform,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& v4_finsub_1(A)
& ~ v1_xboole_0(B)
& v4_finsub_1(B)
& m1_subset_1(C,k2_zfmisc_1(A,B))
& m1_subset_1(D,k2_zfmisc_1(A,B)) )
=> m1_subset_1(k2_normform(A,B,C,D),k2_zfmisc_1(A,B)) ) ).
fof(commutativity_k2_normform,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& v4_finsub_1(A)
& ~ v1_xboole_0(B)
& v4_finsub_1(B)
& m1_subset_1(C,k2_zfmisc_1(A,B))
& m1_subset_1(D,k2_zfmisc_1(A,B)) )
=> k2_normform(A,B,C,D) = k2_normform(A,B,D,C) ) ).
fof(idempotence_k2_normform,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& v4_finsub_1(A)
& ~ v1_xboole_0(B)
& v4_finsub_1(B)
& m1_subset_1(C,k2_zfmisc_1(A,B))
& m1_subset_1(D,k2_zfmisc_1(A,B)) )
=> k2_normform(A,B,C,C) = C ) ).
fof(dt_k3_normform,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& v4_finsub_1(A)
& ~ v1_xboole_0(B)
& v4_finsub_1(B)
& m1_subset_1(C,k2_zfmisc_1(A,B))
& m1_subset_1(D,k2_zfmisc_1(A,B)) )
=> m1_subset_1(k3_normform(A,B,C,D),k2_zfmisc_1(A,B)) ) ).
fof(dt_k4_normform,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& v4_finsub_1(A)
& ~ v1_xboole_0(B)
& v4_finsub_1(B)
& m1_subset_1(C,k2_zfmisc_1(A,B))
& m1_subset_1(D,k2_zfmisc_1(A,B)) )
=> m1_subset_1(k4_normform(A,B,C,D),k2_zfmisc_1(A,B)) ) ).
fof(commutativity_k4_normform,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& v4_finsub_1(A)
& ~ v1_xboole_0(B)
& v4_finsub_1(B)
& m1_subset_1(C,k2_zfmisc_1(A,B))
& m1_subset_1(D,k2_zfmisc_1(A,B)) )
=> k4_normform(A,B,C,D) = k4_normform(A,B,D,C) ) ).
fof(dt_k5_normform,axiom,
! [A] :
( v1_funct_1(k5_normform(A))
& v1_funct_2(k5_normform(A),k2_zfmisc_1(k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A))),k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)))
& m2_relset_1(k5_normform(A),k2_zfmisc_1(k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A))),k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A))) ) ).
fof(dt_k6_normform,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(C,k5_finsub_1(A))
& v1_funct_1(D)
& v1_funct_2(D,A,k2_zfmisc_1(k5_finsub_1(B),k5_finsub_1(B)))
& m1_relset_1(D,A,k2_zfmisc_1(k5_finsub_1(B),k5_finsub_1(B))) )
=> m1_subset_1(k6_normform(A,B,C,D),k2_zfmisc_1(k5_finsub_1(B),k5_finsub_1(B))) ) ).
fof(dt_k7_normform,axiom,
! [A] : m1_subset_1(k7_normform(A),k1_zfmisc_1(k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)))) ).
fof(dt_k8_normform,axiom,
! [A] : m1_subset_1(k8_normform(A),k1_zfmisc_1(k5_finsub_1(k7_normform(A)))) ).
fof(dt_k9_normform,axiom,
! [A,B] :
( m1_subset_1(B,k5_finsub_1(k7_normform(A)))
=> m2_subset_1(k9_normform(A,B),k5_finsub_1(k7_normform(A)),k8_normform(A)) ) ).
fof(dt_k10_normform,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k5_finsub_1(k7_normform(A)))
& m1_subset_1(C,k5_finsub_1(k7_normform(A))) )
=> m1_subset_1(k10_normform(A,B,C),k5_finsub_1(k7_normform(A))) ) ).
fof(dt_k11_normform,axiom,
! [A,B,C,D,E] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(A))
& v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(B,B),B)
& m1_relset_1(C,k2_zfmisc_1(B,B),B)
& m1_subset_1(D,B)
& m1_subset_1(E,B) )
=> m2_subset_1(k11_normform(A,B,C,D,E),A,B) ) ).
fof(redefinition_k11_normform,axiom,
! [A,B,C,D,E] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(A))
& v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(B,B),B)
& m1_relset_1(C,k2_zfmisc_1(B,B),B)
& m1_subset_1(D,B)
& m1_subset_1(E,B) )
=> k11_normform(A,B,C,D,E) = k1_binop_1(C,D,E) ) ).
fof(dt_k12_normform,axiom,
! [A] :
( v3_lattices(k12_normform(A))
& l3_lattices(k12_normform(A)) ) ).
fof(d8_normform,axiom,
! [A] : k7_normform(A) = a_1_0_normform(A) ).
fof(d9_normform,axiom,
! [A] : k8_normform(A) = a_1_1_normform(A) ).
fof(d10_normform,axiom,
! [A,B] :
( m1_subset_1(B,k5_finsub_1(k7_normform(A)))
=> k9_normform(A,B) = a_2_0_normform(A,B) ) ).
fof(d11_normform,axiom,
! [A,B] :
( m1_subset_1(B,k5_finsub_1(k7_normform(A)))
=> ! [C] :
( m1_subset_1(C,k5_finsub_1(k7_normform(A)))
=> k10_normform(A,B,C) = k3_xboole_0(k7_normform(A),a_3_0_normform(A,B,C)) ) ) ).
fof(fraenkel_a_1_0_normform,axiom,
! [A,B] :
( r2_hidden(A,a_1_0_normform(B))
<=> ? [C] :
( m1_subset_1(C,k2_zfmisc_1(k5_finsub_1(B),k5_finsub_1(B)))
& A = C
& r1_xboole_0(k2_domain_1(k5_finsub_1(B),k5_finsub_1(B),C),k3_domain_1(k5_finsub_1(B),k5_finsub_1(B),C)) ) ) ).
fof(fraenkel_a_1_1_normform,axiom,
! [A,B] :
( r2_hidden(A,a_1_1_normform(B))
<=> ? [C] :
( m1_subset_1(C,k5_finsub_1(k7_normform(B)))
& A = C
& ! [D] :
( m2_subset_1(D,k2_zfmisc_1(k5_finsub_1(B),k5_finsub_1(B)),k7_normform(B))
=> ! [E] :
( m2_subset_1(E,k2_zfmisc_1(k5_finsub_1(B),k5_finsub_1(B)),k7_normform(B))
=> ( ( r2_hidden(D,C)
& r2_hidden(E,C)
& r1_normform(k5_finsub_1(B),k5_finsub_1(B),D,E) )
=> D = E ) ) ) ) ) ).
fof(fraenkel_a_2_0_normform,axiom,
! [A,B,C] :
( m1_subset_1(C,k5_finsub_1(k7_normform(B)))
=> ( r2_hidden(A,a_2_0_normform(B,C))
<=> ? [D] :
( m2_subset_1(D,k2_zfmisc_1(k5_finsub_1(B),k5_finsub_1(B)),k7_normform(B))
& A = D
& ! [E] :
( m2_subset_1(E,k2_zfmisc_1(k5_finsub_1(B),k5_finsub_1(B)),k7_normform(B))
=> ( ( r2_hidden(E,C)
& r1_normform(k5_finsub_1(B),k5_finsub_1(B),E,D) )
<=> E = D ) ) ) ) ) ).
fof(fraenkel_a_3_0_normform,axiom,
! [A,B,C,D] :
( ( m1_subset_1(C,k5_finsub_1(k7_normform(B)))
& m1_subset_1(D,k5_finsub_1(k7_normform(B))) )
=> ( r2_hidden(A,a_3_0_normform(B,C,D))
<=> ? [E,F] :
( m2_subset_1(E,k2_zfmisc_1(k5_finsub_1(B),k5_finsub_1(B)),k7_normform(B))
& m2_subset_1(F,k2_zfmisc_1(k5_finsub_1(B),k5_finsub_1(B)),k7_normform(B))
& A = k1_normform(k5_finsub_1(B),k5_finsub_1(B),E,F)
& r2_hidden(E,C)
& r2_hidden(F,D) ) ) ) ).
%------------------------------------------------------------------------------