SET007 Axioms: SET007+401.ax
%------------------------------------------------------------------------------
% File : SET007+401 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Quantales
% 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 : quantal1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 127 ( 5 unt; 0 def)
% Number of atoms : 1115 ( 103 equ)
% Maximal formula atoms : 20 ( 8 avg)
% Number of connectives : 1099 ( 111 ~; 0 |; 695 &)
% ( 43 <=>; 250 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 9 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 56 ( 55 usr; 0 prp; 1-3 aty)
% Number of functors : 67 ( 67 usr; 13 con; 0-6 aty)
% Number of variables : 367 ( 326 !; 41 ?)
% 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_quantal1,axiom,
? [A] :
( l1_quantal1(A)
& v2_quantal1(A) ) ).
fof(rc2_quantal1,axiom,
? [A] :
( l1_quantal1(A)
& ~ v3_struct_0(A) ) ).
fof(rc3_quantal1,axiom,
? [A] :
( l2_quantal1(A)
& v3_quantal1(A) ) ).
fof(rc4_quantal1,axiom,
? [A] :
( l2_quantal1(A)
& ~ v3_struct_0(A) ) ).
fof(cc1_quantal1,axiom,
! [A] :
( l1_group_1(A)
=> ( ( ~ v3_struct_0(A)
& v6_quantal1(A) )
=> ( ~ v3_struct_0(A)
& v4_quantal1(A)
& v5_quantal1(A) ) ) ) ).
fof(cc2_quantal1,axiom,
! [A] :
( l1_group_1(A)
=> ( ( ~ v3_struct_0(A)
& v4_quantal1(A)
& v5_quantal1(A) )
=> ( ~ v3_struct_0(A)
& v6_quantal1(A) ) ) ) ).
fof(rc5_quantal1,axiom,
? [A] :
( l1_group_1(A)
& ~ v3_struct_0(A)
& v4_quantal1(A)
& v5_quantal1(A)
& v6_quantal1(A) ) ).
fof(fc1_quantal1,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m1_relset_1(B,k2_zfmisc_1(A,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(D)
& v1_funct_2(D,k2_zfmisc_1(A,A),A)
& m1_relset_1(D,k2_zfmisc_1(A,A),A) )
=> ( ~ v3_struct_0(g1_quantal1(A,B,C,D))
& v2_quantal1(g1_quantal1(A,B,C,D)) ) ) ).
fof(rc6_quantal1,axiom,
? [A] :
( l1_quantal1(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)
& v4_lattice3(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v4_quantal1(A)
& v5_quantal1(A)
& v6_quantal1(A)
& v7_quantal1(A)
& v8_quantal1(A) ) ).
fof(fc2_quantal1,axiom,
! [A,B,C,D,E] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m1_relset_1(B,k2_zfmisc_1(A,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(D)
& v1_funct_2(D,k2_zfmisc_1(A,A),A)
& m1_relset_1(D,k2_zfmisc_1(A,A),A)
& m1_subset_1(E,A) )
=> ( ~ v3_struct_0(g2_quantal1(A,B,C,D,E))
& v3_quantal1(g2_quantal1(A,B,C,D,E)) ) ) ).
fof(rc7_quantal1,axiom,
? [A] :
( l2_quantal1(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)
& v4_lattice3(A) ) ).
fof(cc3_quantal1,axiom,
! [A] :
( l2_quantal1(A)
=> ( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& v7_quantal1(A)
& v8_quantal1(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)
& v4_lattice3(A)
& v9_quantal1(A)
& v10_quantal1(A) ) ) ) ).
fof(rc8_quantal1,axiom,
? [A] :
( l2_quantal1(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)
& v4_lattice3(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v4_quantal1(A)
& v5_quantal1(A)
& v6_quantal1(A)
& v7_quantal1(A)
& v8_quantal1(A)
& v9_quantal1(A)
& v10_quantal1(A) ) ).
fof(rc9_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& l3_lattices(A) )
=> ? [B] :
( m1_relset_1(B,u1_struct_0(A),u1_struct_0(A))
& v1_relat_1(B)
& v1_funct_1(B)
& ~ v1_xboole_0(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v1_partfun1(B,u1_struct_0(A),u1_struct_0(A))
& v12_quantal1(B,A)
& v13_quantal1(B,A)
& v14_quantal1(B,A) ) ) ).
fof(rc10_quantal1,axiom,
? [A] :
( l3_quantal1(A)
& v19_quantal1(A) ) ).
fof(rc11_quantal1,axiom,
? [A] :
( l3_quantal1(A)
& ~ v3_struct_0(A) ) ).
fof(fc3_quantal1,axiom,
! [A,B,C,D,E,F] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m1_relset_1(B,k2_zfmisc_1(A,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(D)
& v1_funct_2(D,k2_zfmisc_1(A,A),A)
& m1_relset_1(D,k2_zfmisc_1(A,A),A)
& m1_subset_1(E,A)
& m1_subset_1(F,A) )
=> ( ~ v3_struct_0(g3_quantal1(A,B,C,D,E,F))
& v19_quantal1(g3_quantal1(A,B,C,D,E,F)) ) ) ).
fof(rc12_quantal1,axiom,
? [A] :
( l3_quantal1(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)
& v4_lattice3(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_quantal1(A)
& v8_quantal1(A)
& v9_quantal1(A)
& v10_quantal1(A)
& v19_quantal1(A)
& v20_quantal1(A)
& v21_quantal1(A) ) ).
fof(t1_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l3_lattices(B) )
=> ( g3_lattices(u1_struct_0(A),u2_lattices(A),u1_lattices(A)) = g3_lattices(u1_struct_0(B),u2_lattices(B),u1_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))
=> ( ( C = E
& D = F )
=> ( k1_lattices(A,C,D) = k1_lattices(B,E,F)
& k2_lattices(A,C,D) = k2_lattices(B,E,F)
& ( r1_lattices(A,C,D)
=> r1_lattices(B,E,F) )
& ( r1_lattices(B,E,F)
=> r1_lattices(A,C,D) ) ) ) ) ) ) ) ) ) ) ).
fof(t2_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_lattices(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l3_lattices(B) )
=> ( g3_lattices(u1_struct_0(A),u2_lattices(A),u1_lattices(A)) = g3_lattices(u1_struct_0(B),u2_lattices(B),u1_lattices(B))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( C = D
=> ( ( r3_lattice3(A,C,E)
=> r3_lattice3(B,D,E) )
& ( r3_lattice3(B,D,E)
=> r3_lattice3(A,C,E) )
& ( r4_lattice3(A,C,E)
=> r4_lattice3(B,D,E) )
& ( r4_lattice3(B,D,E)
=> r4_lattice3(A,C,E) ) ) ) ) ) ) ) ) ).
fof(d1_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v1_quantal1(B,A)
<=> ! [C] :
( ( v1_finset_1(C)
& m1_subset_1(C,k1_zfmisc_1(B)) )
=> ? [D] :
( m1_subset_1(D,u1_struct_0(A))
& r1_lattices(A,k15_lattice3(A,C),D)
& r2_hidden(D,B) ) ) ) ) ) ).
fof(t3_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v1_quantal1(B,A)
& v1_xboole_0(B) ) ) ) ).
fof(d2_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ( v4_quantal1(A)
<=> ? [B] :
( m1_subset_1(B,u1_struct_0(A))
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k1_group_1(A,B,C) = B ) ) ) ) ).
fof(d3_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ( v5_quantal1(A)
<=> ? [B] :
( m1_subset_1(B,u1_struct_0(A))
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k1_group_1(A,C,B) = B ) ) ) ) ).
fof(d4_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ( v6_quantal1(A)
<=> ( v4_quantal1(A)
& v5_quantal1(A) ) ) ) ).
fof(d7_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_quantal1(A) )
=> ( v9_quantal1(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_monoid_0(A,k1_lattices(A,B,C),D) = k1_lattices(A,k4_monoid_0(A,B,D),k4_monoid_0(A,C,D))
& k4_monoid_0(A,D,k1_lattices(A,B,C)) = k1_lattices(A,k4_monoid_0(A,D,B),k4_monoid_0(A,D,C)) ) ) ) ) ) ) ).
fof(t4_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_quantal1(A) )
=> ( g3_lattices(u1_struct_0(A),u2_lattices(A),u1_lattices(A)) = k1_lattice3(k1_xboole_0)
=> ( v4_group_1(A)
& v7_group_1(A)
& v2_group_1(A)
& v6_quantal1(A)
& v4_lattice3(A)
& v7_quantal1(A)
& v8_quantal1(A)
& v10_lattices(A) ) ) ) ).
fof(t6_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& v7_quantal1(A)
& v8_quantal1(A)
& l1_quantal1(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_monoid_0(A,k3_lattices(A,B,C),D) = k3_lattices(A,k4_monoid_0(A,B,D),k4_monoid_0(A,C,D))
& k4_monoid_0(A,D,k3_lattices(A,B,C)) = k3_lattices(A,k4_monoid_0(A,D,B),k4_monoid_0(A,D,C)) ) ) ) ) ) ).
fof(t7_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& l2_quantal1(A) )
=> ( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& v4_group_1(A)
& v7_quantal1(A)
& v8_quantal1(A)
& l1_quantal1(A) )
=> ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& v2_group_1(A)
& v4_group_1(A)
& v4_quantal1(A)
& v6_quantal1(A)
& v9_quantal1(A)
& v10_quantal1(A)
& l2_quantal1(A) ) ) ) ).
fof(t8_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& v4_group_1(A)
& v7_quantal1(A)
& v8_quantal1(A)
& l1_quantal1(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,B,C)
=> ( r3_lattices(A,k4_monoid_0(A,B,D),k4_monoid_0(A,C,D))
& r3_lattices(A,k4_monoid_0(A,D,B),k4_monoid_0(A,D,C)) ) ) ) ) ) ) ).
fof(t9_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& v4_group_1(A)
& v7_quantal1(A)
& v8_quantal1(A)
& l1_quantal1(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))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( ( r3_lattices(A,B,C)
& r3_lattices(A,D,E) )
=> r3_lattices(A,k4_monoid_0(A,B,D),k4_monoid_0(A,C,E)) ) ) ) ) ) ) ).
fof(d9_quantal1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v11_quantal1(A)
<=> k5_relat_1(A,A) = A ) ) ).
fof(d10_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(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)) )
=> ( v12_quantal1(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> r1_lattices(A,C,k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,C)) ) ) ) ) ).
fof(d11_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(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)) )
=> ( v13_quantal1(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> r1_lattices(A,k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,C),C) ) ) ) ) ).
fof(d12_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(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] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r1_lattices(A,C,D)
=> r1_lattices(A,k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,C),k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,D)) ) ) ) ) ) ) ).
fof(d14_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_quantal1(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)) )
=> ( v16_quantal1(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> r1_lattices(A,k4_monoid_0(A,k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,C),k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,D)),k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,k4_monoid_0(A,C,D))) ) ) ) ) ) ).
fof(t11_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& v4_group_1(A)
& v7_quantal1(A)
& v8_quantal1(A)
& l1_quantal1(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_monoid_0(A,B,C),D)
<=> r3_lattices(A,C,k2_quantal1(A,B,D)) ) ) ) ) ) ).
fof(t12_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& v4_group_1(A)
& v7_quantal1(A)
& v8_quantal1(A)
& l1_quantal1(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_monoid_0(A,B,C),D)
<=> r3_lattices(A,B,k1_quantal1(A,C,D)) ) ) ) ) ) ).
fof(t13_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& v4_group_1(A)
& v7_quantal1(A)
& v8_quantal1(A)
& l1_quantal1(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,C,D)
=> ( r3_lattices(A,k1_quantal1(A,D,B),k1_quantal1(A,C,B))
& r3_lattices(A,k2_quantal1(A,D,B),k2_quantal1(A,C,B)) ) ) ) ) ) ) ).
fof(t14_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& v4_group_1(A)
& v7_quantal1(A)
& v8_quantal1(A)
& l1_quantal1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(A)) )
=> ( ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k8_funct_2(u1_struct_0(A),u1_struct_0(A),C,D) = k1_quantal1(A,k1_quantal1(A,D,B),B) )
=> v14_quantal1(C,A) ) ) ) ) ).
fof(d17_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_quantal1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( v17_quantal1(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( k2_quantal1(A,k1_quantal1(A,C,B),B) = C
& k1_quantal1(A,k2_quantal1(A,C,B),B) = C ) ) ) ) ) ).
fof(d18_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_quantal1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( v18_quantal1(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k1_quantal1(A,C,B) = k2_quantal1(A,C,B) ) ) ) ) ).
fof(t15_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& v4_group_1(A)
& v7_quantal1(A)
& v8_quantal1(A)
& l1_quantal1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( v18_quantal1(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r3_lattices(A,k4_monoid_0(A,C,D),B)
=> r3_lattices(A,k4_monoid_0(A,D,C),B) ) ) ) ) ) ) ).
fof(t16_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& v4_group_1(A)
& v7_quantal1(A)
& v8_quantal1(A)
& l1_quantal1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( v18_quantal1(B,A)
=> ( r3_lattices(A,C,k1_quantal1(A,k1_quantal1(A,C,B),B))
& r3_lattices(A,C,k2_quantal1(A,k2_quantal1(A,C,B),B)) ) ) ) ) ) ).
fof(t17_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& v4_group_1(A)
& v7_quantal1(A)
& v8_quantal1(A)
& l1_quantal1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( v18_quantal1(B,A)
=> ( k1_quantal1(A,C,B) = k1_quantal1(A,k1_quantal1(A,k1_quantal1(A,C,B),B),B)
& k2_quantal1(A,C,B) = k2_quantal1(A,k2_quantal1(A,k2_quantal1(A,C,B),B),B) ) ) ) ) ) ).
fof(t18_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& v4_group_1(A)
& v7_quantal1(A)
& v8_quantal1(A)
& l1_quantal1(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))
=> ( v18_quantal1(B,A)
=> r3_lattices(A,k4_monoid_0(A,k1_quantal1(A,k1_quantal1(A,C,B),B),k1_quantal1(A,k1_quantal1(A,D,B),B)),k1_quantal1(A,k1_quantal1(A,k4_monoid_0(A,C,D),B),B)) ) ) ) ) ) ).
fof(t19_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& v4_group_1(A)
& v7_quantal1(A)
& v8_quantal1(A)
& l1_quantal1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( v17_quantal1(B,A)
=> ( v2_group_1(A)
& k3_binop_1(u1_struct_0(A),u1_group_1(A)) = k1_quantal1(A,B,B)
& k3_binop_1(u1_struct_0(A),u1_group_1(A)) = k2_quantal1(A,B,B) ) ) ) ) ).
fof(t20_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& v4_group_1(A)
& v7_quantal1(A)
& v8_quantal1(A)
& l1_quantal1(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))
=> ( v17_quantal1(B,A)
=> ( k1_quantal1(A,C,D) = k1_quantal1(A,k4_monoid_0(A,C,k2_quantal1(A,D,B)),B)
& k2_quantal1(A,C,D) = k2_quantal1(A,k4_monoid_0(A,k1_quantal1(A,D,B),C),B) ) ) ) ) ) ) ).
fof(d19_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_quantal1(A) )
=> ( v20_quantal1(A)
<=> v18_quantal1(u1_quantal1(A),A) ) ) ).
fof(d20_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_quantal1(A) )
=> ( v21_quantal1(A)
<=> v17_quantal1(u1_quantal1(A),A) ) ) ).
fof(t21_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_quantal1(A) )
=> ( g3_lattices(u1_struct_0(A),u2_lattices(A),u1_lattices(A)) = k1_lattice3(k1_xboole_0)
=> ( v20_quantal1(A)
& v21_quantal1(A) ) ) ) ).
fof(d21_quantal1,axiom,
! [A] :
( l3_quantal1(A)
=> k3_quantal1(A) = u1_quantal1(A) ) ).
fof(d22_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_quantal1(A) )
=> k4_quantal1(A) = k1_quantal1(A,k3_quantal1(A),k3_quantal1(A)) ) ).
fof(d23_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_quantal1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k5_quantal1(A,B) = k1_quantal1(A,B,k3_quantal1(A)) ) ) ).
fof(d24_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_quantal1(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)) )
=> ( B = k6_quantal1(A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,C) = k5_quantal1(A,C) ) ) ) ) ).
fof(d25_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_quantal1(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)) )
=> k7_quantal1(A,B) = k2_monoid_0(u1_struct_0(A),B,k6_quantal1(A)) ) ) ).
fof(d26_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_quantal1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A))
& m2_relset_1(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A)) )
=> k8_quantal1(A,B) = k7_funct_2(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A),u1_struct_0(A),B,k6_quantal1(A)) ) ) ).
fof(t22_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_quantal1(A)
& v8_quantal1(A)
& v20_quantal1(A)
& v21_quantal1(A)
& l3_quantal1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k5_quantal1(A,k5_quantal1(A,B)) = B ) ) ).
fof(t23_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_quantal1(A)
& v8_quantal1(A)
& v20_quantal1(A)
& v21_quantal1(A)
& l3_quantal1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r3_lattices(A,B,C)
=> r3_lattices(A,k5_quantal1(A,C),k5_quantal1(A,B)) ) ) ) ) ).
fof(t26_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_quantal1(A)
& v8_quantal1(A)
& v20_quantal1(A)
& v21_quantal1(A)
& l3_quantal1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( k5_quantal1(A,k3_lattices(A,B,C)) = k4_lattices(A,k5_quantal1(A,B),k5_quantal1(A,C))
& k5_quantal1(A,k4_lattices(A,B,C)) = k3_lattices(A,k5_quantal1(A,B),k5_quantal1(A,C)) ) ) ) ) ).
fof(d27_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_quantal1(A)
& v8_quantal1(A)
& v20_quantal1(A)
& v21_quantal1(A)
& l3_quantal1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k9_quantal1(A,B,C) = k5_quantal1(A,k4_monoid_0(A,k5_quantal1(A,B),k5_quantal1(A,C))) ) ) ) ).
fof(t29_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_quantal1(A)
& v8_quantal1(A)
& v20_quantal1(A)
& v21_quantal1(A)
& l3_quantal1(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_quantal1(A,B,k4_lattices(A,C,D)) = k4_lattices(A,k9_quantal1(A,B,C),k9_quantal1(A,B,D))
& k9_quantal1(A,k4_lattices(A,C,D),B) = k4_lattices(A,k9_quantal1(A,C,B),k9_quantal1(A,D,B)) ) ) ) ) ) ).
fof(t30_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_quantal1(A)
& v8_quantal1(A)
& v20_quantal1(A)
& v21_quantal1(A)
& l3_quantal1(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))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( ( r3_lattices(A,B,C)
& r3_lattices(A,D,E) )
=> r3_lattices(A,k9_quantal1(A,B,D),k9_quantal1(A,C,E)) ) ) ) ) ) ) ).
fof(t31_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_quantal1(A)
& v8_quantal1(A)
& v20_quantal1(A)
& v21_quantal1(A)
& l3_quantal1(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_quantal1(A,k9_quantal1(A,B,C),D) = k9_quantal1(A,B,k9_quantal1(A,C,D)) ) ) ) ) ).
fof(t32_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_quantal1(A)
& v8_quantal1(A)
& v20_quantal1(A)
& v21_quantal1(A)
& l3_quantal1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( k4_monoid_0(A,B,k4_quantal1(A)) = B
& k4_monoid_0(A,k4_quantal1(A),B) = B ) ) ) ).
fof(t33_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_quantal1(A)
& v8_quantal1(A)
& v20_quantal1(A)
& v21_quantal1(A)
& l3_quantal1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( k9_quantal1(A,B,k3_quantal1(A)) = B
& k9_quantal1(A,k3_quantal1(A),B) = B ) ) ) ).
fof(t34_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& v4_group_1(A)
& v7_quantal1(A)
& v8_quantal1(A)
& l1_quantal1(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)
& v11_quantal1(B)
& v15_quantal1(B,A)
& ! [C] :
( ( ~ v3_struct_0(C)
& v10_lattices(C)
& v4_lattice3(C)
& l3_lattices(C) )
=> ~ ( u1_struct_0(C) = k2_relat_1(B)
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(C)))
=> k15_lattice3(C,D) = k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,k15_lattice3(A,D)) ) ) ) ) ) ) ).
fof(dt_l1_quantal1,axiom,
! [A] :
( l1_quantal1(A)
=> ( l3_lattices(A)
& l1_group_1(A) ) ) ).
fof(existence_l1_quantal1,axiom,
? [A] : l1_quantal1(A) ).
fof(dt_l2_quantal1,axiom,
! [A] :
( l2_quantal1(A)
=> ( l1_quantal1(A)
& l1_vectsp_1(A) ) ) ).
fof(existence_l2_quantal1,axiom,
? [A] : l2_quantal1(A) ).
fof(dt_l3_quantal1,axiom,
! [A] :
( l3_quantal1(A)
=> l2_quantal1(A) ) ).
fof(existence_l3_quantal1,axiom,
? [A] : l3_quantal1(A) ).
fof(abstractness_v2_quantal1,axiom,
! [A] :
( l1_quantal1(A)
=> ( v2_quantal1(A)
=> A = g1_quantal1(u1_struct_0(A),u2_lattices(A),u1_lattices(A),u1_group_1(A)) ) ) ).
fof(abstractness_v3_quantal1,axiom,
! [A] :
( l2_quantal1(A)
=> ( v3_quantal1(A)
=> A = g2_quantal1(u1_struct_0(A),u2_lattices(A),u1_lattices(A),u1_group_1(A),u1_vectsp_1(A)) ) ) ).
fof(abstractness_v19_quantal1,axiom,
! [A] :
( l3_quantal1(A)
=> ( v19_quantal1(A)
=> A = g3_quantal1(u1_struct_0(A),u2_lattices(A),u1_lattices(A),u1_group_1(A),u1_vectsp_1(A),u1_quantal1(A)) ) ) ).
fof(dt_k1_quantal1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_quantal1(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> m1_subset_1(k1_quantal1(A,B,C),u1_struct_0(A)) ) ).
fof(dt_k2_quantal1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_quantal1(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> m1_subset_1(k2_quantal1(A,B,C),u1_struct_0(A)) ) ).
fof(dt_k3_quantal1,axiom,
! [A] :
( l3_quantal1(A)
=> m1_subset_1(k3_quantal1(A),u1_struct_0(A)) ) ).
fof(dt_k4_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_quantal1(A) )
=> m1_subset_1(k4_quantal1(A),u1_struct_0(A)) ) ).
fof(dt_k5_quantal1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l3_quantal1(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> m1_subset_1(k5_quantal1(A,B),u1_struct_0(A)) ) ).
fof(dt_k6_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_quantal1(A) )
=> ( v1_funct_1(k6_quantal1(A))
& v1_funct_2(k6_quantal1(A),u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(k6_quantal1(A),u1_struct_0(A),u1_struct_0(A)) ) ) ).
fof(dt_k7_quantal1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l3_quantal1(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)) )
=> ( v1_funct_1(k7_quantal1(A,B))
& v1_funct_2(k7_quantal1(A,B),u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(k7_quantal1(A,B),u1_struct_0(A),u1_struct_0(A)) ) ) ).
fof(dt_k8_quantal1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l3_quantal1(A)
& v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A))
& m1_relset_1(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A)) )
=> ( v1_funct_1(k8_quantal1(A,B))
& v1_funct_2(k8_quantal1(A,B),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A))
& m2_relset_1(k8_quantal1(A,B),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A)) ) ) ).
fof(dt_k9_quantal1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_quantal1(A)
& v8_quantal1(A)
& v20_quantal1(A)
& v21_quantal1(A)
& l3_quantal1(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> m1_subset_1(k9_quantal1(A,B,C),u1_struct_0(A)) ) ).
fof(dt_u1_quantal1,axiom,
! [A] :
( l3_quantal1(A)
=> m1_subset_1(u1_quantal1(A),u1_struct_0(A)) ) ).
fof(dt_g1_quantal1,axiom,
! [A,B,C,D] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m1_relset_1(B,k2_zfmisc_1(A,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(D)
& v1_funct_2(D,k2_zfmisc_1(A,A),A)
& m1_relset_1(D,k2_zfmisc_1(A,A),A) )
=> ( v2_quantal1(g1_quantal1(A,B,C,D))
& l1_quantal1(g1_quantal1(A,B,C,D)) ) ) ).
fof(free_g1_quantal1,axiom,
! [A,B,C,D] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m1_relset_1(B,k2_zfmisc_1(A,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(D)
& v1_funct_2(D,k2_zfmisc_1(A,A),A)
& m1_relset_1(D,k2_zfmisc_1(A,A),A) )
=> ! [E,F,G,H] :
( g1_quantal1(A,B,C,D) = g1_quantal1(E,F,G,H)
=> ( A = E
& B = F
& C = G
& D = H ) ) ) ).
fof(dt_g2_quantal1,axiom,
! [A,B,C,D,E] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m1_relset_1(B,k2_zfmisc_1(A,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(D)
& v1_funct_2(D,k2_zfmisc_1(A,A),A)
& m1_relset_1(D,k2_zfmisc_1(A,A),A)
& m1_subset_1(E,A) )
=> ( v3_quantal1(g2_quantal1(A,B,C,D,E))
& l2_quantal1(g2_quantal1(A,B,C,D,E)) ) ) ).
fof(free_g2_quantal1,axiom,
! [A,B,C,D,E] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m1_relset_1(B,k2_zfmisc_1(A,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(D)
& v1_funct_2(D,k2_zfmisc_1(A,A),A)
& m1_relset_1(D,k2_zfmisc_1(A,A),A)
& m1_subset_1(E,A) )
=> ! [F,G,H,I,J] :
( g2_quantal1(A,B,C,D,E) = g2_quantal1(F,G,H,I,J)
=> ( A = F
& B = G
& C = H
& D = I
& E = J ) ) ) ).
fof(dt_g3_quantal1,axiom,
! [A,B,C,D,E,F] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m1_relset_1(B,k2_zfmisc_1(A,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(D)
& v1_funct_2(D,k2_zfmisc_1(A,A),A)
& m1_relset_1(D,k2_zfmisc_1(A,A),A)
& m1_subset_1(E,A)
& m1_subset_1(F,A) )
=> ( v19_quantal1(g3_quantal1(A,B,C,D,E,F))
& l3_quantal1(g3_quantal1(A,B,C,D,E,F)) ) ) ).
fof(free_g3_quantal1,axiom,
! [A,B,C,D,E,F] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m1_relset_1(B,k2_zfmisc_1(A,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(D)
& v1_funct_2(D,k2_zfmisc_1(A,A),A)
& m1_relset_1(D,k2_zfmisc_1(A,A),A)
& m1_subset_1(E,A)
& m1_subset_1(F,A) )
=> ! [G,H,I,J,K,L] :
( g3_quantal1(A,B,C,D,E,F) = g3_quantal1(G,H,I,J,K,L)
=> ( A = G
& B = H
& C = I
& D = J
& E = K
& F = L ) ) ) ).
fof(d5_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_quantal1(A) )
=> ( v7_quantal1(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] : k4_monoid_0(A,B,k15_lattice3(A,C)) = k15_lattice3(A,a_3_0_quantal1(A,B,C)) ) ) ) ).
fof(d6_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_quantal1(A) )
=> ( v8_quantal1(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] : k4_monoid_0(A,k15_lattice3(A,C),B) = k15_lattice3(A,a_3_1_quantal1(A,B,C)) ) ) ) ).
fof(d8_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_quantal1(A) )
=> ( v10_quantal1(A)
<=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v1_quantal1(B,A)
& v1_quantal1(C,A) )
=> k4_monoid_0(A,k15_lattice3(A,B),k15_lattice3(A,C)) = k15_lattice3(A,a_3_2_quantal1(A,B,C)) ) ) ) ) ) ).
fof(t5_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& v7_quantal1(A)
& v8_quantal1(A)
& l1_quantal1(A) )
=> ! [B,C] : k4_monoid_0(A,k15_lattice3(A,B),k15_lattice3(A,C)) = k15_lattice3(A,a_3_3_quantal1(A,B,C)) ) ).
fof(d13_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(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)) )
=> ( v15_quantal1(B,A)
<=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> r1_lattices(A,k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,k15_lattice3(A,C)),k15_lattice3(A,a_3_4_quantal1(A,B,C))) ) ) ) ) ).
fof(t10_quantal1,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)
=> ( v15_quantal1(B,A)
<=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,k15_lattice3(A,C)) = k15_lattice3(A,a_3_5_quantal1(A,B,C)) ) ) ) ) ) ).
fof(d15_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_quantal1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k1_quantal1(A,B,C) = k15_lattice3(A,a_3_6_quantal1(A,B,C)) ) ) ) ).
fof(d16_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_quantal1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k2_quantal1(A,B,C) = k15_lattice3(A,a_3_7_quantal1(A,B,C)) ) ) ) ).
fof(t24_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_quantal1(A)
& v8_quantal1(A)
& v20_quantal1(A)
& v21_quantal1(A)
& l3_quantal1(A) )
=> ! [B] : k5_quantal1(A,k15_lattice3(A,B)) = k16_lattice3(A,a_2_0_quantal1(A,B)) ) ).
fof(t25_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_quantal1(A)
& v8_quantal1(A)
& v20_quantal1(A)
& v21_quantal1(A)
& l3_quantal1(A) )
=> ! [B] : k5_quantal1(A,k16_lattice3(A,B)) = k15_lattice3(A,a_2_0_quantal1(A,B)) ) ).
fof(t27_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_quantal1(A)
& v8_quantal1(A)
& v20_quantal1(A)
& v21_quantal1(A)
& l3_quantal1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( k4_monoid_0(A,B,k15_lattice3(A,C)) = k15_lattice3(A,a_3_8_quantal1(A,B,C))
& k9_quantal1(A,B,k16_lattice3(A,C)) = k16_lattice3(A,a_3_9_quantal1(A,B,C)) ) ) ) ).
fof(t28_quantal1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v4_lattice3(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_quantal1(A)
& v8_quantal1(A)
& v20_quantal1(A)
& v21_quantal1(A)
& l3_quantal1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( k4_monoid_0(A,k15_lattice3(A,C),B) = k15_lattice3(A,a_3_10_quantal1(A,B,C))
& k9_quantal1(A,k16_lattice3(A,C),B) = k16_lattice3(A,a_3_11_quantal1(A,B,C)) ) ) ) ).
fof(s1_quantal1,axiom,
a_0_0_quantal1 = a_0_2_quantal1 ).
fof(s2_quantal1,axiom,
( ! [A] :
( m1_subset_1(A,f1_s2_quantal1)
=> ~ p1_s2_quantal1(A) )
=> a_0_3_quantal1 = k1_xboole_0 ) ).
fof(s3_quantal1,axiom,
k15_lattice3(f1_s3_quantal1,a_0_4_quantal1) = k15_lattice3(f1_s3_quantal1,a_0_5_quantal1) ).
fof(fraenkel_a_3_0_quantal1,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& l1_quantal1(B)
& m1_subset_1(C,u1_struct_0(B)) )
=> ( r2_hidden(A,a_3_0_quantal1(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(B))
& A = k4_monoid_0(B,C,E)
& r2_hidden(E,D) ) ) ) ).
fof(fraenkel_a_3_1_quantal1,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& l1_quantal1(B)
& m1_subset_1(C,u1_struct_0(B)) )
=> ( r2_hidden(A,a_3_1_quantal1(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(B))
& A = k4_monoid_0(B,E,C)
& r2_hidden(E,D) ) ) ) ).
fof(fraenkel_a_3_2_quantal1,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& l1_quantal1(B)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) )
=> ( r2_hidden(A,a_3_2_quantal1(B,C,D))
<=> ? [E,F] :
( m1_subset_1(E,u1_struct_0(B))
& m1_subset_1(F,u1_struct_0(B))
& A = k4_monoid_0(B,E,F)
& r2_hidden(E,C)
& r2_hidden(F,D) ) ) ) ).
fof(fraenkel_a_3_3_quantal1,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& v4_lattice3(B)
& v7_quantal1(B)
& v8_quantal1(B)
& l1_quantal1(B) )
=> ( r2_hidden(A,a_3_3_quantal1(B,C,D))
<=> ? [E,F] :
( m1_subset_1(E,u1_struct_0(B))
& m1_subset_1(F,u1_struct_0(B))
& A = k4_monoid_0(B,E,F)
& r2_hidden(E,C)
& r2_hidden(F,D) ) ) ) ).
fof(fraenkel_a_3_4_quantal1,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(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))
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) )
=> ( r2_hidden(A,a_3_4_quantal1(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(B))
& A = k8_funct_2(u1_struct_0(B),u1_struct_0(B),C,E)
& r2_hidden(E,D) ) ) ) ).
fof(fraenkel_a_3_5_quantal1,axiom,
! [A,B,C,D] :
( ( ~ 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))
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) )
=> ( r2_hidden(A,a_3_5_quantal1(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(B))
& A = k8_funct_2(u1_struct_0(B),u1_struct_0(B),C,E)
& r2_hidden(E,D) ) ) ) ).
fof(fraenkel_a_3_6_quantal1,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& l1_quantal1(B)
& m1_subset_1(C,u1_struct_0(B))
& m1_subset_1(D,u1_struct_0(B)) )
=> ( r2_hidden(A,a_3_6_quantal1(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(B))
& A = E
& r1_lattices(B,k4_monoid_0(B,E,C),D) ) ) ) ).
fof(fraenkel_a_3_7_quantal1,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& l1_quantal1(B)
& m1_subset_1(C,u1_struct_0(B))
& m1_subset_1(D,u1_struct_0(B)) )
=> ( r2_hidden(A,a_3_7_quantal1(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(B))
& A = E
& r1_lattices(B,k4_monoid_0(B,C,E),D) ) ) ) ).
fof(fraenkel_a_2_0_quantal1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& v4_lattice3(B)
& v2_group_1(B)
& v4_group_1(B)
& v7_quantal1(B)
& v8_quantal1(B)
& v20_quantal1(B)
& v21_quantal1(B)
& l3_quantal1(B) )
=> ( r2_hidden(A,a_2_0_quantal1(B,C))
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(B))
& A = k5_quantal1(B,D)
& r2_hidden(D,C) ) ) ) ).
fof(fraenkel_a_3_8_quantal1,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& v4_lattice3(B)
& v2_group_1(B)
& v4_group_1(B)
& v7_quantal1(B)
& v8_quantal1(B)
& v20_quantal1(B)
& v21_quantal1(B)
& l3_quantal1(B)
& m1_subset_1(C,u1_struct_0(B)) )
=> ( r2_hidden(A,a_3_8_quantal1(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(B))
& A = k4_monoid_0(B,C,E)
& r2_hidden(E,D) ) ) ) ).
fof(fraenkel_a_3_9_quantal1,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& v4_lattice3(B)
& v2_group_1(B)
& v4_group_1(B)
& v7_quantal1(B)
& v8_quantal1(B)
& v20_quantal1(B)
& v21_quantal1(B)
& l3_quantal1(B)
& m1_subset_1(C,u1_struct_0(B)) )
=> ( r2_hidden(A,a_3_9_quantal1(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(B))
& A = k9_quantal1(B,C,E)
& r2_hidden(E,D) ) ) ) ).
fof(fraenkel_a_3_10_quantal1,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& v4_lattice3(B)
& v2_group_1(B)
& v4_group_1(B)
& v7_quantal1(B)
& v8_quantal1(B)
& v20_quantal1(B)
& v21_quantal1(B)
& l3_quantal1(B)
& m1_subset_1(C,u1_struct_0(B)) )
=> ( r2_hidden(A,a_3_10_quantal1(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(B))
& A = k4_monoid_0(B,E,C)
& r2_hidden(E,D) ) ) ) ).
fof(fraenkel_a_3_11_quantal1,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& v10_lattices(B)
& v4_lattice3(B)
& v2_group_1(B)
& v4_group_1(B)
& v7_quantal1(B)
& v8_quantal1(B)
& v20_quantal1(B)
& v21_quantal1(B)
& l3_quantal1(B)
& m1_subset_1(C,u1_struct_0(B)) )
=> ( r2_hidden(A,a_3_11_quantal1(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(B))
& A = k9_quantal1(B,E,C)
& r2_hidden(E,D) ) ) ) ).
fof(fraenkel_a_0_0_quantal1,axiom,
! [A] :
( r2_hidden(A,a_0_0_quantal1)
<=> ? [B] :
( m1_subset_1(B,f2_s1_quantal1)
& A = f3_s1_quantal1(B)
& r2_hidden(B,a_0_1_quantal1) ) ) ).
fof(fraenkel_a_0_1_quantal1,axiom,
! [A] :
( r2_hidden(A,a_0_1_quantal1)
<=> ? [B] :
( m1_subset_1(B,f1_s1_quantal1)
& A = f4_s1_quantal1(B)
& p1_s1_quantal1(B) ) ) ).
fof(fraenkel_a_0_2_quantal1,axiom,
! [A] :
( r2_hidden(A,a_0_2_quantal1)
<=> ? [B] :
( m1_subset_1(B,f1_s1_quantal1)
& A = f3_s1_quantal1(f4_s1_quantal1(B))
& p1_s1_quantal1(B) ) ) ).
fof(fraenkel_a_0_3_quantal1,axiom,
! [A] :
( r2_hidden(A,a_0_3_quantal1)
<=> ? [B] :
( m1_subset_1(B,f1_s2_quantal1)
& A = f2_s2_quantal1(B)
& p1_s2_quantal1(B) ) ) ).
fof(fraenkel_a_0_4_quantal1,axiom,
! [A] :
( r2_hidden(A,a_0_4_quantal1)
<=> ? [B] :
( m1_subset_1(B,u1_struct_0(f1_s3_quantal1))
& A = k15_lattice3(f1_s3_quantal1,a_1_0_quantal1(B))
& r2_hidden(B,f3_s3_quantal1) ) ) ).
fof(fraenkel_a_1_0_quantal1,axiom,
! [A,B] :
( m1_subset_1(B,u1_struct_0(f1_s3_quantal1))
=> ( r2_hidden(A,a_1_0_quantal1(B))
<=> ? [C] :
( m1_subset_1(C,u1_struct_0(f1_s3_quantal1))
& A = f2_s3_quantal1(B,C)
& r2_hidden(C,f4_s3_quantal1) ) ) ) ).
fof(fraenkel_a_0_5_quantal1,axiom,
! [A] :
( r2_hidden(A,a_0_5_quantal1)
<=> ? [B,C] :
( m1_subset_1(B,u1_struct_0(f1_s3_quantal1))
& m1_subset_1(C,u1_struct_0(f1_s3_quantal1))
& A = f2_s3_quantal1(B,C)
& r2_hidden(B,f3_s3_quantal1)
& r2_hidden(C,f4_s3_quantal1) ) ) ).
%------------------------------------------------------------------------------