SET007 Axioms: SET007+461.ax
%------------------------------------------------------------------------------
% File : SET007+461 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : More on the Lattice of Congruences in Many Sorted Algebra
% 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 : msualg_8 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 34 ( 3 unt; 0 def)
% Number of atoms : 294 ( 29 equ)
% Maximal formula atoms : 20 ( 8 avg)
% Number of connectives : 305 ( 45 ~; 2 |; 168 &)
% ( 7 <=>; 83 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 10 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 43 ( 42 usr; 0 prp; 1-4 aty)
% Number of functors : 35 ( 35 usr; 4 con; 0-5 aty)
% Number of variables : 112 ( 105 !; 7 ?)
% 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_msualg_8,axiom,
! [A] :
( ~ v3_struct_0(k2_msualg_5(A))
& v3_lattices(k2_msualg_5(A))
& v4_lattices(k2_msualg_5(A))
& v5_lattices(k2_msualg_5(A))
& v6_lattices(k2_msualg_5(A))
& v7_lattices(k2_msualg_5(A))
& v8_lattices(k2_msualg_5(A))
& v9_lattices(k2_msualg_5(A))
& v10_lattices(k2_msualg_5(A))
& v13_lattices(k2_msualg_5(A))
& v14_lattices(k2_msualg_5(A))
& v15_lattices(k2_msualg_5(A)) ) ).
fof(fc2_msualg_8,axiom,
! [A] :
( ~ v3_struct_0(k2_msualg_5(A))
& v3_lattices(k2_msualg_5(A))
& v4_lattices(k2_msualg_5(A))
& v5_lattices(k2_msualg_5(A))
& v6_lattices(k2_msualg_5(A))
& v7_lattices(k2_msualg_5(A))
& v8_lattices(k2_msualg_5(A))
& v9_lattices(k2_msualg_5(A))
& v10_lattices(k2_msualg_5(A))
& v13_lattices(k2_msualg_5(A))
& v14_lattices(k2_msualg_5(A))
& v15_lattices(k2_msualg_5(A))
& v4_lattice3(k2_msualg_5(A)) ) ).
fof(fc3_msualg_8,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ( ~ v3_struct_0(k6_msualg_5(A,B))
& v3_lattices(k6_msualg_5(A,B))
& v4_lattices(k6_msualg_5(A,B))
& v5_lattices(k6_msualg_5(A,B))
& v6_lattices(k6_msualg_5(A,B))
& v7_lattices(k6_msualg_5(A,B))
& v8_lattices(k6_msualg_5(A,B))
& v9_lattices(k6_msualg_5(A,B))
& v10_lattices(k6_msualg_5(A,B))
& v13_lattices(k6_msualg_5(A,B))
& v14_lattices(k6_msualg_5(A,B))
& v15_lattices(k6_msualg_5(A,B))
& v4_lattice3(k6_msualg_5(A,B))
& v1_msualg_7(k6_msualg_5(A,B),k5_msualg_5(u1_struct_0(A),u4_msualg_1(A,B)))
& v2_msualg_7(k6_msualg_5(A,B),k5_msualg_5(u1_struct_0(A),u4_msualg_1(A,B))) ) ) ).
fof(t1_msualg_8,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ( ( r1_xreal_0(np__1,A)
=> ( r1_xreal_0(k3_finseq_1(B),A)
| ( r2_hidden(A,k4_finseq_1(B))
& r2_hidden(k1_nat_1(A,np__1),k4_finseq_1(B)) ) ) )
& ( ( r2_hidden(A,k4_finseq_1(B))
& r2_hidden(k1_nat_1(A,np__1),k4_finseq_1(B)) )
=> ( r1_xreal_0(np__1,A)
& ~ r1_xreal_0(k3_finseq_1(B),A) ) ) ) ) ) ).
fof(t2_msualg_8,axiom,
! [A,B] :
( m1_subset_1(B,u1_struct_0(k2_msualg_5(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k2_msualg_5(A)))
=> ! [D] :
( ( v3_relat_2(D)
& v8_relat_2(D)
& v1_partfun1(D,A,A)
& m2_relset_1(D,A,A) )
=> ! [E] :
( ( v3_relat_2(E)
& v8_relat_2(E)
& v1_partfun1(E,A,A)
& m2_relset_1(E,A,A) )
=> ( ( B = D
& C = E )
=> ( r3_lattices(k2_msualg_5(A),B,C)
<=> r1_tarski(D,E) ) ) ) ) ) ) ).
fof(t3_msualg_8,axiom,
! [A] : k5_lattices(k2_msualg_5(A)) = k6_partfun1(A) ).
fof(t4_msualg_8,axiom,
! [A] : k6_lattices(k2_msualg_5(A)) = k1_eqrel_1(A) ).
fof(t5_msualg_8,axiom,
! [A] : v4_lattice3(k2_msualg_5(A)) ).
fof(t6_msualg_8,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k2_msualg_5(A))))
=> m2_relset_1(k3_tarski(B),A,A) ) ).
fof(t7_msualg_8,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k2_msualg_5(A))))
=> r1_tarski(k3_tarski(B),k15_lattice3(k2_msualg_5(A),B)) ) ).
fof(t8_msualg_8,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k2_msualg_5(A))))
=> ! [C] :
( m2_relset_1(C,A,A)
=> ( C = k3_tarski(B)
=> k15_lattice3(k2_msualg_5(A),B) = k1_msualg_5(A,C) ) ) ) ).
fof(t9_msualg_8,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k2_msualg_5(A))))
=> ! [C] :
( v1_relat_1(C)
=> ( C = k3_tarski(B)
=> C = k4_relat_1(C) ) ) ) ).
fof(t10_msualg_8,axiom,
! [A,B,C,D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k2_msualg_5(C))))
=> ( ( r2_hidden(A,C)
& r2_hidden(B,C) )
=> ( r2_hidden(k4_tarski(A,B),k15_lattice3(k2_msualg_5(C),D))
<=> ? [E] :
( v1_relat_1(E)
& v1_funct_1(E)
& v1_finseq_1(E)
& r1_xreal_0(np__1,k3_finseq_1(E))
& A = k1_funct_1(E,np__1)
& B = k1_funct_1(E,k3_finseq_1(E))
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,F)
=> ( r1_xreal_0(k3_finseq_1(E),F)
| r2_hidden(k4_tarski(k1_funct_1(E,F),k1_funct_1(E,k1_nat_1(F,np__1))),k3_tarski(D)) ) ) ) ) ) ) ) ).
fof(t11_msualg_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_msualg_5(A,B))))
=> ( v3_msualg_4(k16_lattice3(k5_msualg_5(u1_struct_0(A),u4_msualg_1(A,B)),C),A,B)
& v4_msualg_4(k16_lattice3(k5_msualg_5(u1_struct_0(A),u4_msualg_1(A,B)),C),A,B)
& m1_msualg_4(k16_lattice3(k5_msualg_5(u1_struct_0(A),u4_msualg_1(A,B)),C),u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B)) ) ) ) ) ).
fof(t12_msualg_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k5_msualg_5(u1_struct_0(A),u4_msualg_1(A,B))))
=> ( ( v3_msualg_4(C,A,B)
& v4_msualg_4(C,A,B)
& m1_msualg_4(C,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B)) )
=> k1_msualg_8(A,B,C) = C ) ) ) ) ).
fof(t13_msualg_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_msualg_5(u1_struct_0(A),u4_msualg_1(A,B)))))
=> r6_pboole(u1_struct_0(A),k1_msualg_8(A,B,k15_lattice3(k5_msualg_5(u1_struct_0(A),u4_msualg_1(A,B)),C)),k2_msualg_8(A,B,C)) ) ) ) ).
fof(t14_msualg_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_msualg_5(A,B))))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k6_msualg_5(A,B))))
=> ! [E] :
( ( v3_msualg_4(E,A,B)
& v4_msualg_4(E,A,B)
& m1_msualg_4(E,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B)) )
=> ! [F] :
( ( v3_msualg_4(F,A,B)
& v4_msualg_4(F,A,B)
& m1_msualg_4(F,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B)) )
=> ( ( E = k15_lattice3(k5_msualg_5(u1_struct_0(A),u4_msualg_1(A,B)),C)
& F = k15_lattice3(k5_msualg_5(u1_struct_0(A),u4_msualg_1(A,B)),D) )
=> k4_msualg_5(u1_struct_0(A),u4_msualg_1(A,B),E,F) = k15_lattice3(k5_msualg_5(u1_struct_0(A),u4_msualg_1(A,B)),k4_subset_1(u1_struct_0(k6_msualg_5(A,B)),C,D)) ) ) ) ) ) ) ) ).
fof(t16_msualg_8,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( ( v3_relat_2(D)
& v8_relat_2(D)
& v1_partfun1(D,k1_funct_1(B,C),k1_funct_1(B,C))
& m2_relset_1(D,k1_funct_1(B,C),k1_funct_1(B,C)) )
=> ? [E] :
( v2_msualg_4(E,A,B)
& m1_msualg_4(E,A,B,B)
& k1_msualg_4(A,B,B,E,C) = D
& ! [F] :
( m1_subset_1(F,A)
=> ( F != C
=> k1_msualg_4(A,B,B,E,F) = k1_eqrel_1(k1_funct_1(B,F)) ) ) ) ) ) ) ) ).
fof(d3_msualg_8,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k5_msualg_5(A,B))))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k2_msualg_5(k1_funct_1(B,C)))))
=> ( E = k3_msualg_8(A,B,C,D)
<=> ! [F] :
( r2_hidden(F,E)
<=> ? [G] :
( v2_msualg_4(G,A,B)
& m1_msualg_4(G,A,B,B)
& F = k1_msualg_4(A,B,B,G,C)
& r2_hidden(G,D) ) ) ) ) ) ) ) ) ).
fof(t17_msualg_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k5_msualg_5(u1_struct_0(A),u4_msualg_1(A,B)))))
=> ! [E] :
( ( v2_msualg_4(E,u1_struct_0(A),u4_msualg_1(A,B))
& m1_msualg_4(E,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B)) )
=> ( E = k15_lattice3(k5_msualg_5(u1_struct_0(A),u4_msualg_1(A,B)),D)
=> k1_msualg_4(u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B),E,C) = k15_lattice3(k2_msualg_5(k1_funct_1(u4_msualg_1(A,B),C)),k3_msualg_8(u1_struct_0(A),u4_msualg_1(A,B),C,D)) ) ) ) ) ) ) ).
fof(t18_msualg_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_msualg_5(A,B))))
=> ( v3_msualg_4(k15_lattice3(k5_msualg_5(u1_struct_0(A),u4_msualg_1(A,B)),C),A,B)
& v4_msualg_4(k15_lattice3(k5_msualg_5(u1_struct_0(A),u4_msualg_1(A,B)),C),A,B)
& m1_msualg_4(k15_lattice3(k5_msualg_5(u1_struct_0(A),u4_msualg_1(A,B)),C),u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B)) ) ) ) ) ).
fof(t19_msualg_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> v1_msualg_7(k6_msualg_5(A,B),k5_msualg_5(u1_struct_0(A),u4_msualg_1(A,B))) ) ) ).
fof(t20_msualg_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> v2_msualg_7(k6_msualg_5(A,B),k5_msualg_5(u1_struct_0(A),u4_msualg_1(A,B))) ) ) ).
fof(s1_msualg_8,axiom,
( ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(A,k2_finseq_1(f1_s1_msualg_8))
& ! [B] : ~ p1_s1_msualg_8(A,B) ) )
=> ? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A)
& k4_finseq_1(A) = k2_finseq_1(f1_s1_msualg_8)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r2_hidden(B,k2_finseq_1(f1_s1_msualg_8))
=> p1_s1_msualg_8(B,k1_funct_1(A,B)) ) ) ) ) ).
fof(dt_k1_msualg_8,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& v5_msualg_1(B,A)
& l3_msualg_1(B,A)
& m1_subset_1(C,u1_struct_0(k5_msualg_5(u1_struct_0(A),u4_msualg_1(A,B)))) )
=> ( v3_msualg_4(k1_msualg_8(A,B,C),A,B)
& v4_msualg_4(k1_msualg_8(A,B,C),A,B)
& m1_msualg_4(k1_msualg_8(A,B,C),u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B)) ) ) ).
fof(dt_k2_msualg_8,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& v5_msualg_1(B,A)
& l3_msualg_1(B,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_msualg_5(u1_struct_0(A),u4_msualg_1(A,B))))) )
=> ( v3_msualg_4(k2_msualg_8(A,B,C),A,B)
& v4_msualg_4(k2_msualg_8(A,B,C),A,B)
& m1_msualg_4(k2_msualg_8(A,B,C),u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B)) ) ) ).
fof(dt_k3_msualg_8,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& m1_pboole(B,A)
& m1_subset_1(C,A)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k5_msualg_5(A,B)))) )
=> m1_subset_1(k3_msualg_8(A,B,C,D),k1_zfmisc_1(u1_struct_0(k2_msualg_5(k1_funct_1(B,C))))) ) ).
fof(redefinition_k3_msualg_8,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& m1_pboole(B,A)
& m1_subset_1(C,A)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k5_msualg_5(A,B)))) )
=> k3_msualg_8(A,B,C,D) = k5_card_3(C,D) ) ).
fof(d1_msualg_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k5_msualg_5(u1_struct_0(A),u4_msualg_1(A,B))))
=> k1_msualg_8(A,B,C) = k16_lattice3(k5_msualg_5(u1_struct_0(A),u4_msualg_1(A,B)),a_3_0_msualg_8(A,B,C)) ) ) ) ).
fof(d2_msualg_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_msualg_5(u1_struct_0(A),u4_msualg_1(A,B)))))
=> k2_msualg_8(A,B,C) = k16_lattice3(k5_msualg_5(u1_struct_0(A),u4_msualg_1(A,B)),a_3_1_msualg_8(A,B,C)) ) ) ) ).
fof(t15_msualg_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_msualg_5(A,B))))
=> k15_lattice3(k5_msualg_5(u1_struct_0(A),u4_msualg_1(A,B)),C) = k15_lattice3(k5_msualg_5(u1_struct_0(A),u4_msualg_1(A,B)),a_3_2_msualg_8(A,B,C)) ) ) ) ).
fof(fraenkel_a_3_0_msualg_8,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B)
& v5_msualg_1(C,B)
& l3_msualg_1(C,B)
& m1_subset_1(D,u1_struct_0(k5_msualg_5(u1_struct_0(B),u4_msualg_1(B,C)))) )
=> ( r2_hidden(A,a_3_0_msualg_8(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(k5_msualg_5(u1_struct_0(B),u4_msualg_1(B,C))))
& A = E
& v3_msualg_4(E,B,C)
& v4_msualg_4(E,B,C)
& m1_msualg_4(E,u1_struct_0(B),u4_msualg_1(B,C),u4_msualg_1(B,C))
& r3_lattices(k5_msualg_5(u1_struct_0(B),u4_msualg_1(B,C)),D,E) ) ) ) ).
fof(fraenkel_a_3_1_msualg_8,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B)
& v5_msualg_1(C,B)
& l3_msualg_1(C,B)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k5_msualg_5(u1_struct_0(B),u4_msualg_1(B,C))))) )
=> ( r2_hidden(A,a_3_1_msualg_8(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(k5_msualg_5(u1_struct_0(B),u4_msualg_1(B,C))))
& A = E
& v3_msualg_4(E,B,C)
& v4_msualg_4(E,B,C)
& m1_msualg_4(E,u1_struct_0(B),u4_msualg_1(B,C),u4_msualg_1(B,C))
& r4_lattice3(k5_msualg_5(u1_struct_0(B),u4_msualg_1(B,C)),E,D) ) ) ) ).
fof(fraenkel_a_3_2_msualg_8,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B)
& v5_msualg_1(C,B)
& l3_msualg_1(C,B)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k6_msualg_5(B,C)))) )
=> ( r2_hidden(A,a_3_2_msualg_8(B,C,D))
<=> ? [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k5_msualg_5(u1_struct_0(B),u4_msualg_1(B,C)))))
& A = k15_lattice3(k5_msualg_5(u1_struct_0(B),u4_msualg_1(B,C)),E)
& v1_finset_1(E)
& m1_subset_1(E,k1_zfmisc_1(D)) ) ) ) ).
%------------------------------------------------------------------------------