SET007 Axioms: SET007+665.ax
%------------------------------------------------------------------------------
% File : SET007+665 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The Incompleteness of the Lattice of Substitutions
% 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 : heyting3 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 64 ( 5 unt; 0 def)
% Number of atoms : 305 ( 39 equ)
% Maximal formula atoms : 16 ( 4 avg)
% Number of connectives : 294 ( 53 ~; 2 |; 99 &)
% ( 12 <=>; 128 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 8 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 42 ( 40 usr; 1 prp; 0-3 aty)
% Number of functors : 43 ( 43 usr; 8 con; 0-4 aty)
% Number of variables : 165 ( 159 !; 6 ?)
% 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_heyting3,axiom,
! [A,B] : v1_funct_1(k2_zfmisc_1(A,k1_tarski(B))) ).
fof(fc2_heyting3,axiom,
! [A,B] :
( ~ v3_struct_0(k1_heyting3(A,B))
& v1_lattice3(k1_heyting3(A,B))
& v2_lattice3(k1_heyting3(A,B)) ) ).
fof(fc3_heyting3,axiom,
! [A,B] :
( ~ v3_struct_0(k1_heyting3(A,B))
& v2_orders_2(k1_heyting3(A,B))
& v3_orders_2(k1_heyting3(A,B))
& v4_orders_2(k1_heyting3(A,B))
& v1_lattice3(k1_heyting3(A,B))
& v2_lattice3(k1_heyting3(A,B)) ) ).
fof(fc4_heyting3,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> v1_finset_1(k2_heyting3(A,B)) ) ).
fof(fc5_heyting3,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> v1_finset_1(k3_heyting3(A,B)) ) ).
fof(fc6_heyting3,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> ( ~ v1_xboole_0(k2_heyting3(A,B))
& v1_finset_1(k2_heyting3(A,B)) ) ) ).
fof(fc7_heyting3,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( ~ v1_xboole_0(k4_heyting3(np__1,A))
& v1_finset_1(k4_heyting3(np__1,A)) ) ) ).
fof(rc1_heyting3,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ? [B] :
( m1_subset_1(B,u1_struct_0(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A))))
& ~ v1_xboole_0(B) ) ) ).
fof(fc8_heyting3,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( ~ v3_struct_0(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A)))
& v2_orders_2(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A)))
& v3_orders_2(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A)))
& v4_orders_2(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A)))
& v1_lattice3(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A)))
& v2_lattice3(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A)))
& ~ v3_lattice3(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A))) ) ) ).
fof(fc9_heyting3,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( ~ v3_struct_0(k5_substlat(k5_numbers,k6_domain_1(k5_numbers,A)))
& v3_lattices(k5_substlat(k5_numbers,k6_domain_1(k5_numbers,A)))
& v4_lattices(k5_substlat(k5_numbers,k6_domain_1(k5_numbers,A)))
& v5_lattices(k5_substlat(k5_numbers,k6_domain_1(k5_numbers,A)))
& v6_lattices(k5_substlat(k5_numbers,k6_domain_1(k5_numbers,A)))
& v7_lattices(k5_substlat(k5_numbers,k6_domain_1(k5_numbers,A)))
& v8_lattices(k5_substlat(k5_numbers,k6_domain_1(k5_numbers,A)))
& v9_lattices(k5_substlat(k5_numbers,k6_domain_1(k5_numbers,A)))
& v10_lattices(k5_substlat(k5_numbers,k6_domain_1(k5_numbers,A)))
& v11_lattices(k5_substlat(k5_numbers,k6_domain_1(k5_numbers,A)))
& v12_lattices(k5_substlat(k5_numbers,k6_domain_1(k5_numbers,A)))
& v13_lattices(k5_substlat(k5_numbers,k6_domain_1(k5_numbers,A)))
& v14_lattices(k5_substlat(k5_numbers,k6_domain_1(k5_numbers,A)))
& v15_lattices(k5_substlat(k5_numbers,k6_domain_1(k5_numbers,A)))
& ~ v4_lattice3(k5_substlat(k5_numbers,k6_domain_1(k5_numbers,A))) ) ) ).
fof(t1_heyting3,axiom,
! [A] :
( ( ~ v1_abian(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> r1_xreal_0(np__1,A) ) ).
fof(t2_heyting3,axiom,
$true ).
fof(t3_heyting3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& m1_subset_1(A,k1_zfmisc_1(k5_numbers)) )
=> ? [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
& r1_tarski(A,k1_finsub_1(k1_zfmisc_1(k5_numbers),k2_finseq_1(B),k6_domain_1(k5_numbers,np__0))) ) ) ).
fof(t4_heyting3,axiom,
! [A] :
( ( v1_finset_1(A)
& m1_subset_1(A,k1_zfmisc_1(k5_numbers)) )
=> ~ ! [B] :
( ( ~ v1_abian(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> r2_hidden(B,A) ) ) ).
fof(t5_heyting3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_finset_1(B)
& m1_subset_1(B,k1_zfmisc_1(k12_mcart_1(k1_numbers,k5_numbers,k5_numbers,k6_domain_1(k5_numbers,A)))) )
=> ? [C] :
( ~ v1_xboole_0(C)
& m2_subset_1(C,k1_numbers,k5_numbers)
& r1_tarski(B,k12_mcart_1(k5_numbers,k5_numbers,k1_finsub_1(k1_zfmisc_1(k5_numbers),k2_finseq_1(C),k6_domain_1(k5_numbers,np__0)),k6_domain_1(k5_numbers,A))) ) ) ) ).
fof(t6_heyting3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_finset_1(B)
& m1_subset_1(B,k1_zfmisc_1(k12_mcart_1(k1_numbers,k5_numbers,k5_numbers,k6_domain_1(k5_numbers,A)))) )
=> ~ ! [C] :
( ( ~ v1_xboole_0(C)
& m2_subset_1(C,k1_numbers,k5_numbers) )
=> r2_hidden(k1_domain_1(k5_numbers,k5_numbers,k1_nat_1(k2_nat_1(np__2,C),np__1),A),B) ) ) ) ).
fof(t7_heyting3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_finset_1(B)
& m1_subset_1(B,k1_zfmisc_1(k12_mcart_1(k1_numbers,k5_numbers,k5_numbers,k6_domain_1(k5_numbers,A)))) )
=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(C,D)
& r2_hidden(k1_domain_1(k5_numbers,k5_numbers,D,A),B) ) ) ) ) ) ).
fof(t8_heyting3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v14_lattices(A)
& l3_lattices(A) )
=> k6_lattices(A) = k4_yellow_0(k3_lattice3(A)) ) ).
fof(t9_heyting3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v13_lattices(A)
& l3_lattices(A) )
=> k5_lattices(A) = k3_yellow_0(k3_lattice3(A)) ) ).
fof(t10_heyting3,axiom,
$true ).
fof(t11_heyting3,axiom,
! [A,B] :
( v1_finset_1(B)
=> ! [C] :
( m1_subset_1(C,k5_finsub_1(k4_partfun1(A,B)))
=> ! [D] :
( m1_subset_1(D,k5_finsub_1(k4_partfun1(A,B)))
=> ( C = k1_xboole_0
=> ( D = k1_xboole_0
| k3_heyting2(A,B,D,C) = k1_xboole_0 ) ) ) ) ) ).
fof(t12_heyting3,axiom,
! [A,B,C,D] :
( ( r1_tarski(A,B)
& r1_tarski(C,D) )
=> r1_tarski(k1_substlat(A,C),k1_substlat(B,D)) ) ).
fof(t13_heyting3,axiom,
! [A,B,C,D,E] :
( m1_subset_1(E,k5_finsub_1(k4_partfun1(A,C)))
=> ! [F] :
( m1_subset_1(F,k5_finsub_1(k4_partfun1(B,D)))
=> ( ( r1_tarski(A,B)
& r1_tarski(C,D)
& E = F )
=> k3_substlat(A,C,E) = k3_substlat(B,D,F) ) ) ) ).
fof(t14_heyting3,axiom,
! [A,B,C,D] :
( ( r1_tarski(A,B)
& r1_tarski(C,D) )
=> u2_lattices(k5_substlat(A,C)) = k1_realset1(u2_lattices(k5_substlat(B,D)),u1_struct_0(k5_substlat(A,C))) ) ).
fof(d1_heyting3,axiom,
! [A,B] : k1_heyting3(A,B) = k3_lattice3(k5_substlat(A,B)) ).
fof(t15_heyting3,axiom,
! [A,B,C] :
( m1_subset_1(C,u1_struct_0(k1_heyting3(A,B)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k1_heyting3(A,B)))
=> ( r3_orders_2(k1_heyting3(A,B),C,D)
<=> ! [E] :
~ ( r2_hidden(E,C)
& ! [F] :
~ ( r2_hidden(F,D)
& r1_tarski(F,E) ) ) ) ) ) ).
fof(t16_heyting3,axiom,
! [A,B,C,D] :
( ( r1_tarski(A,B)
& r1_tarski(C,D) )
=> ( v4_yellow_0(k1_heyting3(A,C),k1_heyting3(B,D))
& m1_yellow_0(k1_heyting3(A,C),k1_heyting3(B,D)) ) ) ).
fof(d2_heyting3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m1_subset_1(C,k4_partfun1(k5_numbers,k6_domain_1(k5_numbers,B)))
=> ( C = k2_heyting3(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> ~ ( ! [E] :
( ( ~ v1_abian(E)
& m2_subset_1(E,k1_numbers,k5_numbers) )
=> ~ ( r1_xreal_0(E,k2_nat_1(np__2,A))
& k1_domain_1(k5_numbers,k5_numbers,E,B) = D ) )
& k1_domain_1(k5_numbers,k5_numbers,k2_nat_1(np__2,A),B) != D ) ) ) ) ) ) ).
fof(d3_heyting3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m1_subset_1(C,k4_partfun1(k5_numbers,k6_domain_1(k5_numbers,B)))
=> ( C = k3_heyting3(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> ? [E] :
( ~ v1_abian(E)
& m2_subset_1(E,k1_numbers,k5_numbers)
& r1_xreal_0(E,k1_nat_1(k2_nat_1(np__2,A),np__1))
& k1_domain_1(k5_numbers,k5_numbers,E,B) = D ) ) ) ) ) ) ).
fof(t17_heyting3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> r2_hidden(k1_domain_1(k5_numbers,k5_numbers,k1_nat_1(k2_nat_1(np__2,A),np__1),B),k3_heyting3(A,B)) ) ) ).
fof(t18_heyting3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> r1_xboole_0(k3_heyting3(A,B),k6_domain_1(k2_zfmisc_1(k5_numbers,k5_numbers),k1_domain_1(k5_numbers,k5_numbers,k1_nat_1(k2_nat_1(np__2,A),np__3),B))) ) ) ).
fof(t19_heyting3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k3_heyting3(k1_nat_1(A,np__1),B) = k2_xboole_0(k3_heyting3(A,B),k6_domain_1(k2_zfmisc_1(k5_numbers,k5_numbers),k1_domain_1(k5_numbers,k5_numbers,k1_nat_1(k2_nat_1(np__2,A),np__3),B))) ) ) ).
fof(t20_heyting3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> r2_xboole_0(k3_heyting3(A,B),k3_heyting3(k1_nat_1(A,np__1),B)) ) ) ).
fof(t21_heyting3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ r1_tarski(k2_heyting3(A,B),k3_heyting3(C,B)) ) ) ) ).
fof(t22_heyting3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(A,C)
=> r1_tarski(k3_heyting3(A,B),k3_heyting3(C,B)) ) ) ) ) ).
fof(t23_heyting3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k2_heyting3(np__1,A) = k7_domain_1(k2_zfmisc_1(k5_numbers,k5_numbers),k1_domain_1(k5_numbers,k5_numbers,np__1,A),k1_domain_1(k5_numbers,k5_numbers,np__2,A)) ) ).
fof(d4_heyting3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m1_subset_1(C,k5_finsub_1(k4_partfun1(k5_numbers,k6_domain_1(k5_numbers,B))))
=> ( C = k4_heyting3(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> ~ ( ! [E] :
( ( ~ v1_xboole_0(E)
& m2_subset_1(E,k1_numbers,k5_numbers) )
=> ~ ( r1_xreal_0(E,A)
& D = k2_heyting3(E,B) ) )
& D != k3_heyting3(A,B) ) ) ) ) ) ) ).
fof(t24_heyting3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
~ ( r2_hidden(C,k4_heyting3(k1_nat_1(A,np__1),B))
& ! [D] :
~ ( r2_hidden(D,k4_heyting3(A,B))
& r1_tarski(D,C) ) ) ) ) ).
fof(t25_heyting3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ r2_hidden(k3_heyting3(A,B),k4_heyting3(k1_nat_1(A,np__1),B)) ) ) ).
fof(t26_heyting3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_tarski(k2_heyting3(A,B),k2_heyting3(C,B))
=> A = C ) ) ) ) ).
fof(t27_heyting3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_tarski(k3_heyting3(A,B),k2_heyting3(C,B))
<=> ~ r1_xreal_0(C,A) ) ) ) ) ).
fof(t28_heyting3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> m1_subset_1(k4_heyting3(A,B),u1_struct_0(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,B)))) ) ) ).
fof(d5_heyting3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A)))))
=> ( B = k5_heyting3(A)
<=> ! [C] :
( r2_hidden(C,B)
<=> ? [D] :
( ~ v1_xboole_0(D)
& m2_subset_1(D,k1_numbers,k5_numbers)
& C = k4_heyting3(D,A) ) ) ) ) ) ).
fof(t29_heyting3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k4_heyting3(np__1,A) = k7_domain_1(k4_partfun1(k5_numbers,k6_domain_1(k5_numbers,A)),k2_heyting3(np__1,A),k3_heyting3(np__1,A)) ) ).
fof(t30_heyting3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k4_heyting3(np__1,A) != k1_tarski(k1_xboole_0) ) ).
fof(t31_heyting3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> m1_subset_1(k6_domain_1(k4_partfun1(k5_numbers,k6_domain_1(k5_numbers,B)),k2_heyting3(A,B)),u1_struct_0(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,B)))) ) ) ).
fof(t32_heyting3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B,C,D] :
( m1_subset_1(D,u1_struct_0(k1_heyting3(B,k6_domain_1(k5_numbers,A))))
=> ( r2_hidden(C,D)
=> ( v1_finset_1(C)
& m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(B,k6_domain_1(k5_numbers,A)))) ) ) ) ) ).
fof(t33_heyting3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A))))
=> ( r1_lattice3(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A)),k5_heyting3(A),B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ~ ( r2_hidden(C,B)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(k1_domain_1(k5_numbers,k5_numbers,D,A),C)
& v1_abian(D) ) ) ) ) ) ) ) ).
fof(t34_heyting3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A))))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A))))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A)))))
=> ( ( r1_lattice3(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A)),D,B)
& r1_lattice3(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A)),D,C) )
=> r1_lattice3(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A)),D,k13_lattice3(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A)),B,C)) ) ) ) ) ) ).
fof(t35_heyting3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A))))
=> ( r2_hidden(k1_xboole_0,B)
=> B = k1_tarski(k1_xboole_0) ) ) ) ).
fof(t36_heyting3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,u1_struct_0(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A)))) )
=> ~ ( B != k1_tarski(k1_xboole_0)
& ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finset_1(C) )
=> ~ ( r2_hidden(C,B)
& C != k1_xboole_0 ) ) ) ) ) ).
fof(t37_heyting3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,u1_struct_0(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A)))) )
=> ! [C] :
( m1_subset_1(C,k5_finsub_1(k4_partfun1(k5_numbers,k6_domain_1(k5_numbers,A))))
=> ( B = C
=> ( B = k1_tarski(k1_xboole_0)
| ( ~ v1_xboole_0(k1_heyting2(k5_numbers,k6_domain_1(k5_numbers,A),C))
& v1_finset_1(k1_heyting2(k5_numbers,k6_domain_1(k5_numbers,A),C))
& m1_subset_1(k1_heyting2(k5_numbers,k6_domain_1(k5_numbers,A),C),k1_zfmisc_1(k5_numbers)) ) ) ) ) ) ) ).
fof(t38_heyting3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A))))
=> ! [C] :
( m1_subset_1(C,k5_finsub_1(k4_partfun1(k5_numbers,k6_domain_1(k5_numbers,A))))
=> ! [D] :
( ( ~ v1_xboole_0(D)
& v1_finset_1(D)
& m1_subset_1(D,k1_zfmisc_1(k5_numbers)) )
=> ( ( D = k1_heyting2(k5_numbers,k6_domain_1(k5_numbers,A),C)
& C = B )
=> ! [E] :
( r2_hidden(E,B)
=> ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ~ ( ~ r1_xreal_0(F,k2_xcmplx_0(k1_pre_circ(D),np__1))
& r2_hidden(k1_domain_1(k5_numbers,k5_numbers,F,A),E) ) ) ) ) ) ) ) ) ).
fof(t39_heyting3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k4_yellow_0(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A))) = k1_tarski(k1_xboole_0) ) ).
fof(t40_heyting3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k3_yellow_0(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A))) = k1_xboole_0 ) ).
fof(t41_heyting3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A))))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A))))
=> ( ( r3_orders_2(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A)),B,C)
& B = k1_tarski(k1_xboole_0) )
=> C = k1_tarski(k1_xboole_0) ) ) ) ) ).
fof(t42_heyting3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A))))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A))))
=> ( ( r3_orders_2(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A)),B,C)
& C = k1_xboole_0 )
=> B = k1_xboole_0 ) ) ) ) ).
fof(t43_heyting3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A))))
=> ~ ( r1_lattice3(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A)),k5_heyting3(A),B)
& B = k1_tarski(k1_xboole_0) ) ) ) ).
fof(s1_heyting3,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(f1_s1_heyting3)))
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(f1_s1_heyting3)))
=> ( ( ! [C] :
( r2_hidden(C,A)
<=> p1_s1_heyting3(C) )
& ! [C] :
( r2_hidden(C,B)
<=> p1_s1_heyting3(C) ) )
=> A = B ) ) ) ).
fof(dt_k1_heyting3,axiom,
! [A,B] : l1_orders_2(k1_heyting3(A,B)) ).
fof(dt_k2_heyting3,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> m1_subset_1(k2_heyting3(A,B),k4_partfun1(k5_numbers,k6_domain_1(k5_numbers,B))) ) ).
fof(dt_k3_heyting3,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> m1_subset_1(k3_heyting3(A,B),k4_partfun1(k5_numbers,k6_domain_1(k5_numbers,B))) ) ).
fof(dt_k4_heyting3,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> m1_subset_1(k4_heyting3(A,B),k5_finsub_1(k4_partfun1(k5_numbers,k6_domain_1(k5_numbers,B)))) ) ).
fof(dt_k5_heyting3,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> m1_subset_1(k5_heyting3(A),k1_zfmisc_1(u1_struct_0(k1_heyting3(k5_numbers,k6_domain_1(k5_numbers,A))))) ) ).
%------------------------------------------------------------------------------