## SET007 Axioms: SET007+453.ax

```%------------------------------------------------------------------------------
% File     : SET007+453 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Many Sorted Closure Operator and the Many Sorted Closure System
% 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    : closure1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  103 (   1 unit)
%            Number of atoms       :  687 (  28 equality)
%            Maximal formula depth :   21 (   9 average)
%            Number of connectives :  590 (   6 ~  ;   0  |; 273  &)
%                                         (  27 <=>; 284 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   42 (   1 propositional; 0-4 arity)
%            Number of functors    :   32 (   3 constant; 0-5 arity)
%            Number of variables   :  374 (   0 singleton; 362 !;  12 ?)
%            Maximal term depth    :    5 (   1 average)
% 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_closure1,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ? [C] :
( m2_pboole(C,A,k7_mssubfam(A,B))
& v1_relat_1(C)
& v3_relat_1(C)
& v1_funct_1(C)
& v1_pre_circ(C,A) ) ) )).

fof(rc2_closure1,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ? [C] :
( m3_pboole(C,A,k7_mssubfam(A,B),k7_mssubfam(A,B))
& v1_relat_1(C)
& v1_funct_1(C)
& v1_funcop_1(C)
& v1_closure1(C,A,B)
& v2_closure1(C,A,B)
& v3_closure1(C,A,B)
& v4_closure1(C,A,B) ) ) )).

fof(cc1_closure1,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m3_pboole(C,A,k7_mssubfam(A,B),k7_mssubfam(A,B))
=> ( v4_closure1(C,A,B)
=> ( v1_funcop_1(C)
& v2_closure1(C,A,B) ) ) ) ) )).

fof(rc3_closure1,axiom,(
! [A] :
( l1_struct_0(A)
=> ? [B] :
( l1_closure1(B,A)
& v5_closure1(B,A) ) ) )).

fof(fc1_closure1,axiom,(
! [A,B] :
( ( l1_struct_0(A)
& l2_msualg_1(B,A) )
=> ( v5_closure1(k5_closure1(A,B),A)
& v6_closure1(k5_closure1(A,B),A)
& v7_closure1(k5_closure1(A,B),A)
& v8_closure1(k5_closure1(A,B),A)
& v9_closure1(k5_closure1(A,B),A)
& v10_closure1(k5_closure1(A,B),A)
& v11_closure1(k5_closure1(A,B),A) ) ) )).

fof(fc2_closure1,axiom,(
! [A,B] :
( ( l1_struct_0(A)
& v5_msualg_1(B,A)
& l2_msualg_1(B,A) )
=> ( v5_msualg_1(k5_closure1(A,B),A)
& v5_closure1(k5_closure1(A,B),A)
& v6_closure1(k5_closure1(A,B),A)
& v7_closure1(k5_closure1(A,B),A)
& v8_closure1(k5_closure1(A,B),A)
& v9_closure1(k5_closure1(A,B),A)
& v10_closure1(k5_closure1(A,B),A)
& v11_closure1(k5_closure1(A,B),A) ) ) )).

fof(rc4_closure1,axiom,(
! [A] :
( l1_struct_0(A)
=> ? [B] :
( l1_closure1(B,A)
& v5_msualg_1(B,A)
& v5_closure1(B,A)
& v6_closure1(B,A)
& v7_closure1(B,A)
& v8_closure1(B,A)
& v9_closure1(B,A)
& v10_closure1(B,A)
& v11_closure1(B,A) ) ) )).

fof(fc3_closure1,axiom,(
! [A,B] :
( ( l1_struct_0(A)
& v6_closure1(B,A)
& l1_closure1(B,A) )
=> ( v1_relat_1(u1_closure1(A,B))
& v1_funct_1(u1_closure1(A,B))
& v1_mssubfam(u1_closure1(A,B),u1_struct_0(A),u4_msualg_1(A,B)) ) ) )).

fof(fc4_closure1,axiom,(
! [A,B] :
( ( l1_struct_0(A)
& v7_closure1(B,A)
& l1_closure1(B,A) )
=> ( v1_relat_1(u1_closure1(A,B))
& v2_relat_1(u1_closure1(A,B))
& v1_funct_1(u1_closure1(A,B))
& v1_mssubfam(u1_closure1(A,B),u1_struct_0(A),u4_msualg_1(A,B))
& v2_mssubfam(u1_closure1(A,B),u1_struct_0(A),u4_msualg_1(A,B))
& v6_mssubfam(u1_closure1(A,B),u1_struct_0(A),u4_msualg_1(A,B)) ) ) )).

fof(fc5_closure1,axiom,(
! [A,B] :
( ( l1_struct_0(A)
& v8_closure1(B,A)
& l1_closure1(B,A) )
=> ( v1_relat_1(u1_closure1(A,B))
& v1_funct_1(u1_closure1(A,B))
& v3_mssubfam(u1_closure1(A,B),u1_struct_0(A),u4_msualg_1(A,B)) ) ) )).

fof(fc6_closure1,axiom,(
! [A,B] :
( ( l1_struct_0(A)
& v9_closure1(B,A)
& l1_closure1(B,A) )
=> ( v1_relat_1(u1_closure1(A,B))
& v2_relat_1(u1_closure1(A,B))
& v1_funct_1(u1_closure1(A,B))
& v3_mssubfam(u1_closure1(A,B),u1_struct_0(A),u4_msualg_1(A,B))
& v4_mssubfam(u1_closure1(A,B),u1_struct_0(A),u4_msualg_1(A,B))
& v5_mssubfam(u1_closure1(A,B),u1_struct_0(A),u4_msualg_1(A,B)) ) ) )).

fof(fc7_closure1,axiom,(
! [A,B] :
( ( l1_struct_0(A)
& v10_closure1(B,A)
& l1_closure1(B,A) )
=> ( v1_relat_1(u1_closure1(A,B))
& v2_relat_1(u1_closure1(A,B))
& v1_funct_1(u1_closure1(A,B))
& v5_mssubfam(u1_closure1(A,B),u1_struct_0(A),u4_msualg_1(A,B)) ) ) )).

fof(fc8_closure1,axiom,(
! [A,B] :
( ( l1_struct_0(A)
& v11_closure1(B,A)
& l1_closure1(B,A) )
=> ( v1_relat_1(u1_closure1(A,B))
& v2_relat_1(u1_closure1(A,B))
& v1_funct_1(u1_closure1(A,B))
& v6_mssubfam(u1_closure1(A,B),u1_struct_0(A),u4_msualg_1(A,B)) ) ) )).

fof(fc9_closure1,axiom,(
! [A,B,C] :
( ( l1_struct_0(A)
& v2_relat_1(B)
& m1_pboole(B,u1_struct_0(A))
& m4_pboole(C,u1_struct_0(A),k1_mboolean(u1_struct_0(A),B)) )
=> ( v5_msualg_1(g1_closure1(A,B,C),A)
& v5_closure1(g1_closure1(A,B,C),A) ) ) )).

fof(fc10_closure1,axiom,(
! [A,B,C] :
( ( l1_struct_0(A)
& l2_msualg_1(B,A)
& v1_mssubfam(C,u1_struct_0(A),u4_msualg_1(A,B))
& m4_pboole(C,u1_struct_0(A),k1_mboolean(u1_struct_0(A),u4_msualg_1(A,B))) )
=> ( v5_closure1(g1_closure1(A,u4_msualg_1(A,B),C),A)
& v6_closure1(g1_closure1(A,u4_msualg_1(A,B),C),A) ) ) )).

fof(fc11_closure1,axiom,(
! [A,B,C] :
( ( l1_struct_0(A)
& l2_msualg_1(B,A)
& v2_mssubfam(C,u1_struct_0(A),u4_msualg_1(A,B))
& m4_pboole(C,u1_struct_0(A),k1_mboolean(u1_struct_0(A),u4_msualg_1(A,B))) )
=> ( v5_closure1(g1_closure1(A,u4_msualg_1(A,B),C),A)
& v6_closure1(g1_closure1(A,u4_msualg_1(A,B),C),A)
& v7_closure1(g1_closure1(A,u4_msualg_1(A,B),C),A) ) ) )).

fof(fc12_closure1,axiom,(
! [A,B,C] :
( ( l1_struct_0(A)
& l2_msualg_1(B,A)
& v3_mssubfam(C,u1_struct_0(A),u4_msualg_1(A,B))
& m4_pboole(C,u1_struct_0(A),k1_mboolean(u1_struct_0(A),u4_msualg_1(A,B))) )
=> ( v5_closure1(g1_closure1(A,u4_msualg_1(A,B),C),A)
& v8_closure1(g1_closure1(A,u4_msualg_1(A,B),C),A) ) ) )).

fof(fc13_closure1,axiom,(
! [A,B,C] :
( ( l1_struct_0(A)
& l2_msualg_1(B,A)
& v4_mssubfam(C,u1_struct_0(A),u4_msualg_1(A,B))
& m4_pboole(C,u1_struct_0(A),k1_mboolean(u1_struct_0(A),u4_msualg_1(A,B))) )
=> ( v5_closure1(g1_closure1(A,u4_msualg_1(A,B),C),A)
& v8_closure1(g1_closure1(A,u4_msualg_1(A,B),C),A)
& v9_closure1(g1_closure1(A,u4_msualg_1(A,B),C),A) ) ) )).

fof(fc14_closure1,axiom,(
! [A,B,C] :
( ( l1_struct_0(A)
& l2_msualg_1(B,A)
& v5_mssubfam(C,u1_struct_0(A),u4_msualg_1(A,B))
& m4_pboole(C,u1_struct_0(A),k1_mboolean(u1_struct_0(A),u4_msualg_1(A,B))) )
=> ( v5_closure1(g1_closure1(A,u4_msualg_1(A,B),C),A)
& v10_closure1(g1_closure1(A,u4_msualg_1(A,B),C),A) ) ) )).

fof(fc15_closure1,axiom,(
! [A,B,C] :
( ( l1_struct_0(A)
& l2_msualg_1(B,A)
& v6_mssubfam(C,u1_struct_0(A),u4_msualg_1(A,B))
& m4_pboole(C,u1_struct_0(A),k1_mboolean(u1_struct_0(A),u4_msualg_1(A,B))) )
=> ( v5_closure1(g1_closure1(A,u4_msualg_1(A,B),C),A)
& v11_closure1(g1_closure1(A,u4_msualg_1(A,B),C),A) ) ) )).

fof(cc2_closure1,axiom,(
! [A] :
( l1_struct_0(A)
=> ! [B] :
( l1_closure1(B,A)
=> ( v7_closure1(B,A)
=> v6_closure1(B,A) ) ) ) )).

fof(cc3_closure1,axiom,(
! [A] :
( l1_struct_0(A)
=> ! [B] :
( l1_closure1(B,A)
=> ( v9_closure1(B,A)
=> v8_closure1(B,A) ) ) ) )).

fof(cc4_closure1,axiom,(
! [A] :
( l1_struct_0(A)
=> ! [B] :
( l1_closure1(B,A)
=> ( v9_closure1(B,A)
=> v10_closure1(B,A) ) ) ) )).

fof(cc5_closure1,axiom,(
! [A] :
( l1_struct_0(A)
=> ! [B] :
( l1_closure1(B,A)
=> ( v7_closure1(B,A)
=> v11_closure1(B,A) ) ) ) )).

fof(fc16_closure1,axiom,(
! [A,B,C] :
( ( v3_relat_1(B)
& m1_pboole(B,A)
& m3_pboole(C,A,B,B) )
=> ( v1_relat_1(k6_closure1(A,B,C))
& v3_relat_1(k6_closure1(A,B,C))
& v1_funct_1(k6_closure1(A,B,C))
& v1_pre_circ(k6_closure1(A,B,C),A) ) ) )).

fof(fc17_closure1,axiom,(
! [A,B,C] :
( ( l1_struct_0(A)
& m1_pboole(B,u1_struct_0(A))
& v1_closure1(C,u1_struct_0(A),B)
& v2_closure1(C,u1_struct_0(A),B)
& v3_closure1(C,u1_struct_0(A),B)
& m3_pboole(C,u1_struct_0(A),k7_mssubfam(u1_struct_0(A),B),k7_mssubfam(u1_struct_0(A),B)) )
=> ( v5_closure1(k7_closure1(A,B,C),A)
& v8_closure1(k7_closure1(A,B,C),A)
& v9_closure1(k7_closure1(A,B,C),A)
& v10_closure1(k7_closure1(A,B,C),A) ) ) )).

fof(fc18_closure1,axiom,(
! [A,B,C] :
( ( l1_struct_0(A)
& v2_relat_1(B)
& m1_pboole(B,u1_struct_0(A))
& v1_closure1(C,u1_struct_0(A),B)
& v2_closure1(C,u1_struct_0(A),B)
& v3_closure1(C,u1_struct_0(A),B)
& m3_pboole(C,u1_struct_0(A),k7_mssubfam(u1_struct_0(A),B),k7_mssubfam(u1_struct_0(A),B)) )
=> ( v5_msualg_1(k7_closure1(A,B,C),A)
& v5_closure1(k7_closure1(A,B,C),A)
& v8_closure1(k7_closure1(A,B,C),A)
& v9_closure1(k7_closure1(A,B,C),A)
& v10_closure1(k7_closure1(A,B,C),A) ) ) )).

fof(t2_closure1,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ( ? [C] :
( m1_pboole(C,A)
& r1_pboole(A,C,B) )
=> v2_relat_1(B) ) ) )).

fof(t3_closure1,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ! [D] :
( ( v2_relat_1(D)
& m1_pboole(D,A) )
=> ! [E] :
( m3_pboole(E,A,B,D)
=> ( r1_pboole(A,C,B)
=> r1_pboole(A,k1_closure1(A,E,C),D) ) ) ) ) ) )).

fof(t4_closure1,axiom,(
! [A,B] :
( ( v1_funcop_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( ( v1_funcop_1(C)
& m1_pboole(C,A) )
=> ! [D] :
( m1_pboole(D,A)
=> ( r1_pboole(A,D,k1_mssubfam(A,C))
=> k1_closure1(A,B,k1_closure1(A,C,D)) = k15_pralg_1(k13_pboole(C,B),D) ) ) ) ) )).

fof(t5_closure1,axiom,(
! [A,B] :
( ( v1_funcop_1(B)
& m1_pboole(B,A) )
=> ( v1_msualg_3(B)
=> ! [C] :
( m1_pboole(C,A)
=> ! [D] :
( m1_pboole(D,A)
=> ( ( r1_pboole(A,C,k1_mssubfam(A,B))
& r1_pboole(A,D,k1_mssubfam(A,B))
& r6_pboole(A,k1_closure1(A,B,C),k1_closure1(A,B,D)) )
=> r6_pboole(A,C,D) ) ) ) ) ) )).

fof(t6_closure1,axiom,(
! [A,B] :
( ( v1_funcop_1(B)
& m1_pboole(B,A) )
=> ( ( v2_relat_1(k1_mssubfam(A,B))
& ! [C] :
( m1_pboole(C,A)
=> ! [D] :
( m1_pboole(D,A)
=> ( ( r1_pboole(A,C,k1_mssubfam(A,B))
& r1_pboole(A,D,k1_mssubfam(A,B))
& r6_pboole(A,k1_closure1(A,B,C),k1_closure1(A,B,D)) )
=> r6_pboole(A,C,D) ) ) ) )
=> v1_msualg_3(B) ) ) )).

fof(t7_closure1,axiom,(
! [A,B] :
( ( v2_relat_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( ( v2_relat_1(C)
& m1_pboole(C,A) )
=> ! [D] :
( m3_pboole(D,A,B,C)
=> ! [E] :
( m3_pboole(E,A,B,C)
=> ( ! [F] :
( m1_pboole(F,A)
=> ( r1_pboole(A,F,B)
=> r6_pboole(A,k1_closure1(A,D,F),k1_closure1(A,E,F)) ) )
=> r6_pboole(A,D,E) ) ) ) ) ) )).

fof(d1_closure1,axiom,(
\$true )).

fof(d2_closure1,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m3_pboole(C,A,k7_mssubfam(A,B),k7_mssubfam(A,B))
=> ( v1_closure1(C,A,B)
<=> ! [D] :
( m2_pboole(D,A,k7_mssubfam(A,B))
=> r2_pboole(A,D,k3_closure1(A,B,C,D)) ) ) ) ) )).

fof(d3_closure1,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m3_pboole(C,A,k7_mssubfam(A,B),k7_mssubfam(A,B))
=> ( v2_closure1(C,A,B)
<=> ! [D] :
( m2_pboole(D,A,k7_mssubfam(A,B))
=> ! [E] :
( m2_pboole(E,A,k7_mssubfam(A,B))
=> ( r2_pboole(A,D,E)
=> r2_pboole(A,k3_closure1(A,B,C,D),k3_closure1(A,B,C,E)) ) ) ) ) ) ) )).

fof(d4_closure1,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m3_pboole(C,A,k7_mssubfam(A,B),k7_mssubfam(A,B))
=> ( v3_closure1(C,A,B)
<=> ! [D] :
( m2_pboole(D,A,k7_mssubfam(A,B))
=> r6_pboole(A,k3_closure1(A,B,C,D),k3_closure1(A,B,C,k3_closure1(A,B,C,D))) ) ) ) ) )).

fof(d5_closure1,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m3_pboole(C,A,k7_mssubfam(A,B),k7_mssubfam(A,B))
=> ( v4_closure1(C,A,B)
<=> ! [D] :
( m2_pboole(D,A,k7_mssubfam(A,B))
=> ! [E] :
( m2_pboole(E,A,k7_mssubfam(A,B))
=> r6_pboole(A,k1_closure1(A,C,k2_pboole(A,D,E)),k2_pboole(A,k3_closure1(A,B,C,D),k3_closure1(A,B,C,E))) ) ) ) ) ) )).

fof(t8_closure1,axiom,(
! [A,B] :
( ( v2_relat_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( m2_pboole(C,A,B)
=> r6_pboole(A,C,k2_closure1(A,B,B,k2_msualg_3(A,B),C)) ) ) )).

fof(t9_closure1,axiom,(
! [A,B] :
( ( v2_relat_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( m2_pboole(C,A,B)
=> ! [D] :
( m2_pboole(D,A,B)
=> ( r2_pboole(A,C,D)
=> r2_pboole(A,k2_closure1(A,B,B,k2_msualg_3(A,B),C),k2_closure1(A,B,B,k2_msualg_3(A,B),D)) ) ) ) ) )).

fof(t10_closure1,axiom,(
! [A,B] :
( ( v2_relat_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( m2_pboole(C,A,B)
=> ! [D] :
( m2_pboole(D,A,B)
=> ( m2_pboole(k2_pboole(A,C,D),A,B)
=> r6_pboole(A,k1_closure1(A,k2_msualg_3(A,B),k2_pboole(A,C,D)),k2_pboole(A,k2_closure1(A,B,B,k2_msualg_3(A,B),C),k2_closure1(A,B,B,k2_msualg_3(A,B),D))) ) ) ) ) )).

fof(t11_closure1,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m2_pboole(C,A,k7_mssubfam(A,B))
=> ! [D,E] :
~ ( r2_hidden(D,A)
& r2_hidden(E,k1_funct_1(k3_closure1(A,B,k2_msualg_3(A,k7_mssubfam(A,B)),C),D))
& ! [F] :
( ( v1_pre_circ(F,A)
& m2_pboole(F,A,k7_mssubfam(A,B)) )
=> ~ ( r2_pboole(A,F,C)
& r2_hidden(E,k1_funct_1(k3_closure1(A,B,k2_msualg_3(A,k7_mssubfam(A,B)),F),D)) ) ) ) ) ) )).

fof(t12_closure1,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ( v1_closure1(k2_msualg_3(A,k7_mssubfam(A,B)),A,B)
& m3_pboole(k2_msualg_3(A,k7_mssubfam(A,B)),A,k7_mssubfam(A,B),k7_mssubfam(A,B)) ) ) )).

fof(t13_closure1,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ( v2_closure1(k2_msualg_3(A,k7_mssubfam(A,B)),A,B)
& m3_pboole(k2_msualg_3(A,k7_mssubfam(A,B)),A,k7_mssubfam(A,B),k7_mssubfam(A,B)) ) ) )).

fof(t14_closure1,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ( v3_closure1(k2_msualg_3(A,k7_mssubfam(A,B)),A,B)
& m3_pboole(k2_msualg_3(A,k7_mssubfam(A,B)),A,k7_mssubfam(A,B),k7_mssubfam(A,B)) ) ) )).

fof(t15_closure1,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ( v4_closure1(k2_msualg_3(A,k7_mssubfam(A,B)),A,B)
& m3_pboole(k2_msualg_3(A,k7_mssubfam(A,B)),A,k7_mssubfam(A,B),k7_mssubfam(A,B)) ) ) )).

fof(t16_closure1,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m3_pboole(C,A,k7_mssubfam(A,B),k7_mssubfam(A,B))
=> ! [D] :
( m2_pboole(D,A,k7_mssubfam(A,B))
=> ( ( r6_pboole(A,D,B)
& v1_closure1(C,A,B) )
=> r6_pboole(A,D,k3_closure1(A,B,C,D)) ) ) ) ) )).

fof(t17_closure1,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m3_pboole(C,A,k7_mssubfam(A,B),k7_mssubfam(A,B))
=> ( ( v1_closure1(C,A,B)
& ! [D] :
( m2_pboole(D,A,k7_mssubfam(A,B))
=> r2_pboole(A,k3_closure1(A,B,C,D),D) ) )
=> v3_closure1(C,A,B) ) ) ) )).

fof(t18_closure1,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m3_pboole(C,A,k7_mssubfam(A,B),k7_mssubfam(A,B))
=> ! [D] :
( m2_pboole(D,A,k7_mssubfam(A,B))
=> ! [E] :
( m2_pboole(E,A,k7_mssubfam(A,B))
=> ( v2_closure1(C,A,B)
=> r2_pboole(A,k1_closure1(A,C,k3_pboole(A,D,E)),k3_pboole(A,k3_closure1(A,B,C,D),k3_closure1(A,B,C,E))) ) ) ) ) ) )).

fof(t19_closure1,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m3_pboole(C,A,k7_mssubfam(A,B),k7_mssubfam(A,B))
=> ! [D] :
( m2_pboole(D,A,k7_mssubfam(A,B))
=> ! [E] :
( m2_pboole(E,A,k7_mssubfam(A,B))
=> ( v4_closure1(C,A,B)
=> r2_pboole(A,k4_pboole(A,k3_closure1(A,B,C,D),k3_closure1(A,B,C,E)),k1_closure1(A,C,k4_pboole(A,D,E))) ) ) ) ) ) )).

fof(t20_closure1,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m3_pboole(C,A,k7_mssubfam(A,B),k7_mssubfam(A,B))
=> ! [D] :
( m3_pboole(D,A,k7_mssubfam(A,B),k7_mssubfam(A,B))
=> ( ( v1_closure1(C,A,B)
& v1_closure1(D,A,B) )
=> v1_closure1(k4_closure1(A,B,D,C),A,B) ) ) ) ) )).

fof(t21_closure1,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m3_pboole(C,A,k7_mssubfam(A,B),k7_mssubfam(A,B))
=> ! [D] :
( m3_pboole(D,A,k7_mssubfam(A,B),k7_mssubfam(A,B))
=> ( ( v2_closure1(C,A,B)
& v2_closure1(D,A,B) )
=> v2_closure1(k4_closure1(A,B,D,C),A,B) ) ) ) ) )).

fof(t22_closure1,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m3_pboole(C,A,k7_mssubfam(A,B),k7_mssubfam(A,B))
=> ! [D] :
( m3_pboole(D,A,k7_mssubfam(A,B),k7_mssubfam(A,B))
=> ( ( v3_closure1(C,A,B)
& v3_closure1(D,A,B)
& r6_pboole(A,k4_closure1(A,B,D,C),k4_closure1(A,B,C,D)) )
=> v3_closure1(k4_closure1(A,B,D,C),A,B) ) ) ) ) )).

fof(t23_closure1,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m3_pboole(C,A,k7_mssubfam(A,B),k7_mssubfam(A,B))
=> ! [D] :
( m3_pboole(D,A,k7_mssubfam(A,B),k7_mssubfam(A,B))
=> ( ( v4_closure1(C,A,B)
& v4_closure1(D,A,B) )
=> v4_closure1(k4_closure1(A,B,D,C),A,B) ) ) ) ) )).

fof(t24_closure1,axiom,(
! [A,B,C] :
( m1_pboole(C,B)
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ! [E] :
( m3_pboole(E,B,k7_mssubfam(B,C),k7_mssubfam(B,C))
=> ( ( v1_closure1(E,B,C)
& r2_hidden(A,B)
& D = k1_funct_1(E,A) )
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(k1_funct_1(C,A)))
=> r1_tarski(F,k1_funct_1(D,F)) ) ) ) ) ) )).

fof(t25_closure1,axiom,(
! [A,B,C] :
( m1_pboole(C,B)
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ! [E] :
( m3_pboole(E,B,k7_mssubfam(B,C),k7_mssubfam(B,C))
=> ( ( v2_closure1(E,B,C)
& r2_hidden(A,B)
& D = k1_funct_1(E,A) )
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(k1_funct_1(C,A)))
=> ! [G] :
( m1_subset_1(G,k1_zfmisc_1(k1_funct_1(C,A)))
=> ( r1_tarski(F,G)
=> r1_tarski(k1_funct_1(D,F),k1_funct_1(D,G)) ) ) ) ) ) ) ) )).

fof(t26_closure1,axiom,(
! [A,B,C] :
( m1_pboole(C,B)
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ! [E] :
( m3_pboole(E,B,k7_mssubfam(B,C),k7_mssubfam(B,C))
=> ( ( v3_closure1(E,B,C)
& r2_hidden(A,B)
& D = k1_funct_1(E,A) )
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(k1_funct_1(C,A)))
=> k1_funct_1(D,F) = k1_funct_1(D,k1_funct_1(D,F)) ) ) ) ) ) )).

fof(t27_closure1,axiom,(
! [A,B,C] :
( m1_pboole(C,B)
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ! [E] :
( m3_pboole(E,B,k7_mssubfam(B,C),k7_mssubfam(B,C))
=> ( ( v4_closure1(E,B,C)
& r2_hidden(A,B)
& D = k1_funct_1(E,A) )
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(k1_funct_1(C,A)))
=> ! [G] :
( m1_subset_1(G,k1_zfmisc_1(k1_funct_1(C,A)))
=> k1_funct_1(D,k4_subset_1(k1_funct_1(C,A),F,G)) = k2_xboole_0(k1_funct_1(D,F),k1_funct_1(D,G)) ) ) ) ) ) ) )).

fof(d6_closure1,axiom,(
! [A] :
( l1_struct_0(A)
=> ! [B] :
( l1_closure1(B,A)
=> ( v6_closure1(B,A)
<=> v1_mssubfam(u1_closure1(A,B),u1_struct_0(A),u4_msualg_1(A,B)) ) ) ) )).

fof(d7_closure1,axiom,(
! [A] :
( l1_struct_0(A)
=> ! [B] :
( l1_closure1(B,A)
=> ( v7_closure1(B,A)
<=> v2_mssubfam(u1_closure1(A,B),u1_struct_0(A),u4_msualg_1(A,B)) ) ) ) )).

fof(d8_closure1,axiom,(
! [A] :
( l1_struct_0(A)
=> ! [B] :
( l1_closure1(B,A)
=> ( v8_closure1(B,A)
<=> v3_mssubfam(u1_closure1(A,B),u1_struct_0(A),u4_msualg_1(A,B)) ) ) ) )).

fof(d9_closure1,axiom,(
! [A] :
( l1_struct_0(A)
=> ! [B] :
( l1_closure1(B,A)
=> ( v9_closure1(B,A)
<=> v4_mssubfam(u1_closure1(A,B),u1_struct_0(A),u4_msualg_1(A,B)) ) ) ) )).

fof(d10_closure1,axiom,(
! [A] :
( l1_struct_0(A)
=> ! [B] :
( l1_closure1(B,A)
=> ( v10_closure1(B,A)
<=> v5_mssubfam(u1_closure1(A,B),u1_struct_0(A),u4_msualg_1(A,B)) ) ) ) )).

fof(d11_closure1,axiom,(
! [A] :
( l1_struct_0(A)
=> ! [B] :
( l1_closure1(B,A)
=> ( v11_closure1(B,A)
<=> v6_mssubfam(u1_closure1(A,B),u1_struct_0(A),u4_msualg_1(A,B)) ) ) ) )).

fof(d12_closure1,axiom,(
! [A] :
( l1_struct_0(A)
=> ! [B] :
( l2_msualg_1(B,A)
=> k5_closure1(A,B) = g1_closure1(A,u4_msualg_1(A,B),k7_mssubfam(u1_struct_0(A),u4_msualg_1(A,B))) ) ) )).

fof(d13_closure1,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m3_pboole(C,A,B,B)
=> ! [D] :
( m4_pboole(D,A,B)
=> ( D = k6_closure1(A,B,C)
<=> ! [E,F] :
( r2_hidden(F,A)
=> ( r2_hidden(E,k1_funct_1(D,F))
<=> ? [G] :
( v1_relat_1(G)
& v1_funct_1(G)
& G = k1_funct_1(C,F)
& r2_hidden(E,k1_relat_1(G))
& k1_funct_1(G,E) = E ) ) ) ) ) ) ) )).

fof(t28_closure1,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ! [D] :
( m3_pboole(D,A,B,B)
=> ( ( r1_pboole(A,C,B)
& r6_pboole(A,k1_closure1(A,D,C),C) )
<=> r1_pboole(A,C,k6_closure1(A,B,D)) ) ) ) ) )).

fof(t29_closure1,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> r6_pboole(A,k6_closure1(A,B,k2_msualg_3(A,B)),B) ) )).

fof(t30_closure1,axiom,(
! [A] :
( l1_struct_0(A)
=> ! [B] :
( m1_pboole(B,u1_struct_0(A))
=> ! [C] :
( ( v1_closure1(C,u1_struct_0(A),B)
& v2_closure1(C,u1_struct_0(A),B)
& m3_pboole(C,u1_struct_0(A),k7_mssubfam(u1_struct_0(A),B),k7_mssubfam(u1_struct_0(A),B)) )
=> ! [D] :
( m4_pboole(D,u1_struct_0(A),k1_mboolean(u1_struct_0(A),B))
=> ( r6_pboole(u1_struct_0(A),D,k6_closure1(u1_struct_0(A),k7_mssubfam(u1_struct_0(A),B),C))
=> ( v9_closure1(g1_closure1(A,B,D),A)
& l1_closure1(g1_closure1(A,B,D),A) ) ) ) ) ) ) )).

fof(t31_closure1,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( ( v5_mssubfam(C,A,B)
& m4_pboole(C,A,k1_mboolean(A,B)) )
=> ! [D] :
( m2_pboole(D,A,k7_mssubfam(A,B))
=> ? [E] :
( v2_relat_1(E)
& m4_pboole(E,A,k1_mboolean(A,B))
& ! [F] :
( m1_pboole(F,A)
=> ( r1_pboole(A,F,E)
<=> ( r1_pboole(A,F,C)
& r2_pboole(A,D,F) ) ) ) ) ) ) ) )).

fof(t33_closure1,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( ( v5_mssubfam(C,A,B)
& m4_pboole(C,A,k1_mboolean(A,B)) )
=> ? [D] :
( m3_pboole(D,A,k7_mssubfam(A,B),k7_mssubfam(A,B))
& ! [E] :
( m2_pboole(E,A,k7_mssubfam(A,B))
=> ! [F] :
( ( v2_relat_1(F)
& m4_pboole(F,A,k1_mboolean(A,B)) )
=> ( ! [G] :
( m1_pboole(G,A)
=> ( r1_pboole(A,G,F)
<=> ( r1_pboole(A,G,C)
& r2_pboole(A,E,G) ) ) )
=> r6_pboole(A,k3_closure1(A,B,D,E),k6_mssubfam(A,B,F)) ) ) ) ) ) ) )).

fof(t34_closure1,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( ( v5_mssubfam(C,A,B)
& m4_pboole(C,A,k1_mboolean(A,B)) )
=> ! [D] :
( m2_pboole(D,A,k7_mssubfam(A,B))
=> ! [E] :
( m3_pboole(E,A,k7_mssubfam(A,B),k7_mssubfam(A,B))
=> ( ( r1_pboole(A,D,C)
& ! [F] :
( m2_pboole(F,A,k7_mssubfam(A,B))
=> ! [G] :
( ( v2_relat_1(G)
& m4_pboole(G,A,k1_mboolean(A,B)) )
=> ( ! [H] :
( m1_pboole(H,A)
=> ( r1_pboole(A,H,G)
<=> ( r1_pboole(A,H,C)
& r2_pboole(A,F,H) ) ) )
=> r6_pboole(A,k3_closure1(A,B,E,F),k6_mssubfam(A,B,G)) ) ) ) )
=> r6_pboole(A,k3_closure1(A,B,E,D),D) ) ) ) ) ) )).

fof(t35_closure1,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( ( v4_mssubfam(C,A,B)
& m4_pboole(C,A,k1_mboolean(A,B)) )
=> ! [D] :
( m2_pboole(D,A,k7_mssubfam(A,B))
=> ! [E] :
( m3_pboole(E,A,k7_mssubfam(A,B),k7_mssubfam(A,B))
=> ( ( r6_pboole(A,k3_closure1(A,B,E,D),D)
& ! [F] :
( m2_pboole(F,A,k7_mssubfam(A,B))
=> ! [G] :
( ( v2_relat_1(G)
& m4_pboole(G,A,k1_mboolean(A,B)) )
=> ( ! [H] :
( m1_pboole(H,A)
=> ( r1_pboole(A,H,G)
<=> ( r1_pboole(A,H,C)
& r2_pboole(A,F,H) ) ) )
=> r6_pboole(A,k3_closure1(A,B,E,F),k6_mssubfam(A,B,G)) ) ) ) )
=> r1_pboole(A,D,C) ) ) ) ) ) )).

fof(t36_closure1,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( ( v5_mssubfam(C,A,B)
& m4_pboole(C,A,k1_mboolean(A,B)) )
=> ! [D] :
( m3_pboole(D,A,k7_mssubfam(A,B),k7_mssubfam(A,B))
=> ( ! [E] :
( m2_pboole(E,A,k7_mssubfam(A,B))
=> ! [F] :
( ( v2_relat_1(F)
& m4_pboole(F,A,k1_mboolean(A,B)) )
=> ( ! [G] :
( m1_pboole(G,A)
=> ( r1_pboole(A,G,F)
<=> ( r1_pboole(A,G,C)
& r2_pboole(A,E,G) ) ) )
=> r6_pboole(A,k3_closure1(A,B,D,E),k6_mssubfam(A,B,F)) ) ) )
=> ( v1_closure1(D,A,B)
& v2_closure1(D,A,B) ) ) ) ) ) )).

fof(t37_closure1,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( ( v4_mssubfam(C,A,B)
& m4_pboole(C,A,k1_mboolean(A,B)) )
=> ! [D] :
( m3_pboole(D,A,k7_mssubfam(A,B),k7_mssubfam(A,B))
=> ( ! [E] :
( m2_pboole(E,A,k7_mssubfam(A,B))
=> ! [F] :
( ( v2_relat_1(F)
& m4_pboole(F,A,k1_mboolean(A,B)) )
=> ( ! [G] :
( m1_pboole(G,A)
=> ( r1_pboole(A,G,F)
<=> ( r1_pboole(A,G,C)
& r2_pboole(A,E,G) ) ) )
=> r6_pboole(A,k3_closure1(A,B,D,E),k6_mssubfam(A,B,F)) ) ) )
=> v3_closure1(D,A,B) ) ) ) ) )).

fof(t38_closure1,axiom,(
! [A] :
( l1_struct_0(A)
=> ! [B] :
( ( v9_closure1(B,A)
& l1_closure1(B,A) )
=> ! [C] :
( m3_pboole(C,u1_struct_0(A),k7_mssubfam(u1_struct_0(A),u4_msualg_1(A,B)),k7_mssubfam(u1_struct_0(A),u4_msualg_1(A,B)))
=> ( ! [D] :
( m2_pboole(D,u1_struct_0(A),k7_mssubfam(u1_struct_0(A),u4_msualg_1(A,B)))
=> ! [E] :
( ( v2_relat_1(E)
& m4_pboole(E,u1_struct_0(A),k1_mboolean(u1_struct_0(A),u4_msualg_1(A,B))) )
=> ( ! [F] :
( m1_pboole(F,u1_struct_0(A))
=> ( r1_pboole(u1_struct_0(A),F,E)
<=> ( r1_pboole(u1_struct_0(A),F,u1_closure1(A,B))
& r2_pboole(u1_struct_0(A),D,F) ) ) )
=> r6_pboole(u1_struct_0(A),k3_closure1(u1_struct_0(A),u4_msualg_1(A,B),C,D),k6_mssubfam(u1_struct_0(A),u4_msualg_1(A,B),E)) ) ) )
=> ( v1_closure1(C,u1_struct_0(A),u4_msualg_1(A,B))
& v2_closure1(C,u1_struct_0(A),u4_msualg_1(A,B))
& v3_closure1(C,u1_struct_0(A),u4_msualg_1(A,B))
& m3_pboole(C,u1_struct_0(A),k7_mssubfam(u1_struct_0(A),u4_msualg_1(A,B)),k7_mssubfam(u1_struct_0(A),u4_msualg_1(A,B))) ) ) ) ) ) )).

fof(d14_closure1,axiom,(
! [A] :
( l1_struct_0(A)
=> ! [B] :
( m1_pboole(B,u1_struct_0(A))
=> ! [C] :
( ( v1_closure1(C,u1_struct_0(A),B)
& v2_closure1(C,u1_struct_0(A),B)
& v3_closure1(C,u1_struct_0(A),B)
& m3_pboole(C,u1_struct_0(A),k7_mssubfam(u1_struct_0(A),B),k7_mssubfam(u1_struct_0(A),B)) )
=> ! [D] :
( ( v9_closure1(D,A)
& l1_closure1(D,A) )
=> ( D = k7_closure1(A,B,C)
<=> ? [E] :
( m4_pboole(E,u1_struct_0(A),k1_mboolean(u1_struct_0(A),B))
& r6_pboole(u1_struct_0(A),E,k6_closure1(u1_struct_0(A),k7_mssubfam(u1_struct_0(A),B),C))
& D = g1_closure1(A,B,E) ) ) ) ) ) ) )).

fof(d15_closure1,axiom,(
! [A] :
( l1_struct_0(A)
=> ! [B] :
( ( v9_closure1(B,A)
& l1_closure1(B,A) )
=> ! [C] :
( ( v1_closure1(C,u1_struct_0(A),u4_msualg_1(A,B))
& v2_closure1(C,u1_struct_0(A),u4_msualg_1(A,B))
& v3_closure1(C,u1_struct_0(A),u4_msualg_1(A,B))
& m3_pboole(C,u1_struct_0(A),k7_mssubfam(u1_struct_0(A),u4_msualg_1(A,B)),k7_mssubfam(u1_struct_0(A),u4_msualg_1(A,B))) )
=> ( C = k8_closure1(A,B)
<=> ! [D] :
( m2_pboole(D,u1_struct_0(A),k7_mssubfam(u1_struct_0(A),u4_msualg_1(A,B)))
=> ! [E] :
( ( v2_relat_1(E)
& m4_pboole(E,u1_struct_0(A),k1_mboolean(u1_struct_0(A),u4_msualg_1(A,B))) )
=> ( ! [F] :
( m1_pboole(F,u1_struct_0(A))
=> ( r1_pboole(u1_struct_0(A),F,E)
<=> ( r1_pboole(u1_struct_0(A),F,u1_closure1(A,B))
& r2_pboole(u1_struct_0(A),D,F) ) ) )
=> r6_pboole(u1_struct_0(A),k3_closure1(u1_struct_0(A),u4_msualg_1(A,B),C,D),k6_mssubfam(u1_struct_0(A),u4_msualg_1(A,B),E)) ) ) ) ) ) ) ) )).

fof(t39_closure1,axiom,(
! [A] :
( l1_struct_0(A)
=> ! [B] :
( m1_pboole(B,u1_struct_0(A))
=> ! [C] :
( ( v1_closure1(C,u1_struct_0(A),B)
& v2_closure1(C,u1_struct_0(A),B)
& v3_closure1(C,u1_struct_0(A),B)
& m3_pboole(C,u1_struct_0(A),k7_mssubfam(u1_struct_0(A),B),k7_mssubfam(u1_struct_0(A),B)) )
=> r6_pboole(u1_struct_0(A),k8_closure1(A,k7_closure1(A,B,C)),C) ) ) ) )).

fof(t40_closure1,axiom,(
! [A] :
( l1_struct_0(A)
=> ! [B] :
( ( v9_closure1(B,A)
& l1_closure1(B,A) )
=> k7_closure1(A,u4_msualg_1(A,B),k8_closure1(A,B)) = g1_closure1(A,u4_msualg_1(A,B),u1_closure1(A,B)) ) ) )).

fof(s1_closure1,axiom,
( ! [A] :
( m1_pboole(A,f1_s1_closure1)
=> ( r1_pboole(f1_s1_closure1,A,f2_s1_closure1)
<=> ( r1_pboole(f1_s1_closure1,A,f3_s1_closure1)
& p1_s1_closure1(A) ) ) )
=> r2_pboole(f1_s1_closure1,f2_s1_closure1,f3_s1_closure1) )).

fof(dt_l1_closure1,axiom,(
! [A] :
( l1_struct_0(A)
=> ! [B] :
( l1_closure1(B,A)
=> l2_msualg_1(B,A) ) ) )).

fof(existence_l1_closure1,axiom,(
! [A] :
( l1_struct_0(A)
=> ? [B] : l1_closure1(B,A) ) )).

fof(abstractness_v5_closure1,axiom,(
! [A,B] :
( ( l1_struct_0(A)
& l1_closure1(B,A) )
=> ( v5_closure1(B,A)
=> B = g1_closure1(A,u4_msualg_1(A,B),u1_closure1(A,B)) ) ) )).

fof(dt_k1_closure1,axiom,(
! [A,B,C] :
( ( v1_funcop_1(B)
& m1_pboole(B,A)
& m1_pboole(C,A) )
=> m1_pboole(k1_closure1(A,B,C),A) ) )).

fof(redefinition_k1_closure1,axiom,(
! [A,B,C] :
( ( v1_funcop_1(B)
& m1_pboole(B,A)
& m1_pboole(C,A) )
=> k1_closure1(A,B,C) = k15_pralg_1(B,C) ) )).

fof(dt_k2_closure1,axiom,(
! [A,B,C,D,E] :
( ( v2_relat_1(B)
& m1_pboole(B,A)
& v2_relat_1(C)
& m1_pboole(C,A)
& m3_pboole(D,A,B,C)
& m2_pboole(E,A,B) )
=> m2_pboole(k2_closure1(A,B,C,D,E),A,C) ) )).

fof(redefinition_k2_closure1,axiom,(
! [A,B,C,D,E] :
( ( v2_relat_1(B)
& m1_pboole(B,A)
& v2_relat_1(C)
& m1_pboole(C,A)
& m3_pboole(D,A,B,C)
& m2_pboole(E,A,B) )
=> k2_closure1(A,B,C,D,E) = k15_pralg_1(D,E) ) )).

fof(dt_k3_closure1,axiom,(
! [A,B,C,D] :
( ( m1_pboole(B,A)
& m3_pboole(C,A,k7_mssubfam(A,B),k7_mssubfam(A,B))
& m2_pboole(D,A,k7_mssubfam(A,B)) )
=> m2_pboole(k3_closure1(A,B,C,D),A,k7_mssubfam(A,B)) ) )).

fof(redefinition_k3_closure1,axiom,(
! [A,B,C,D] :
( ( m1_pboole(B,A)
& m3_pboole(C,A,k7_mssubfam(A,B),k7_mssubfam(A,B))
& m2_pboole(D,A,k7_mssubfam(A,B)) )
=> k3_closure1(A,B,C,D) = k15_pralg_1(C,D) ) )).

fof(dt_k4_closure1,axiom,(
! [A,B,C,D] :
( ( m1_pboole(B,A)
& m3_pboole(C,A,k7_mssubfam(A,B),k7_mssubfam(A,B))
& m3_pboole(D,A,k7_mssubfam(A,B),k7_mssubfam(A,B)) )
=> m3_pboole(k4_closure1(A,B,C,D),A,k7_mssubfam(A,B),k7_mssubfam(A,B)) ) )).

fof(redefinition_k4_closure1,axiom,(
! [A,B,C,D] :
( ( m1_pboole(B,A)
& m3_pboole(C,A,k7_mssubfam(A,B),k7_mssubfam(A,B))
& m3_pboole(D,A,k7_mssubfam(A,B),k7_mssubfam(A,B)) )
=> k4_closure1(A,B,C,D) = k13_pboole(C,D) ) )).

fof(dt_k5_closure1,axiom,(
! [A,B] :
( ( l1_struct_0(A)
& l2_msualg_1(B,A) )
=> l1_closure1(k5_closure1(A,B),A) ) )).

fof(dt_k6_closure1,axiom,(
! [A,B,C] :
( ( m1_pboole(B,A)
& m3_pboole(C,A,B,B) )
=> m4_pboole(k6_closure1(A,B,C),A,B) ) )).

fof(dt_k7_closure1,axiom,(
! [A,B,C] :
( ( l1_struct_0(A)
& m1_pboole(B,u1_struct_0(A))
& v1_closure1(C,u1_struct_0(A),B)
& v2_closure1(C,u1_struct_0(A),B)
& v3_closure1(C,u1_struct_0(A),B)
& m3_pboole(C,u1_struct_0(A),k7_mssubfam(u1_struct_0(A),B),k7_mssubfam(u1_struct_0(A),B)) )
=> ( v9_closure1(k7_closure1(A,B,C),A)
& l1_closure1(k7_closure1(A,B,C),A) ) ) )).

fof(dt_k8_closure1,axiom,(
! [A,B] :
( ( l1_struct_0(A)
& v9_closure1(B,A)
& l1_closure1(B,A) )
=> ( v1_closure1(k8_closure1(A,B),u1_struct_0(A),u4_msualg_1(A,B))
& v2_closure1(k8_closure1(A,B),u1_struct_0(A),u4_msualg_1(A,B))
& v3_closure1(k8_closure1(A,B),u1_struct_0(A),u4_msualg_1(A,B))
& m3_pboole(k8_closure1(A,B),u1_struct_0(A),k7_mssubfam(u1_struct_0(A),u4_msualg_1(A,B)),k7_mssubfam(u1_struct_0(A),u4_msualg_1(A,B))) ) ) )).

fof(dt_u1_closure1,axiom,(
! [A,B] :
( ( l1_struct_0(A)
& l1_closure1(B,A) )
=> m4_pboole(u1_closure1(A,B),u1_struct_0(A),k1_mboolean(u1_struct_0(A),u4_msualg_1(A,B))) ) )).

fof(dt_g1_closure1,axiom,(
! [A,B,C] :
( ( l1_struct_0(A)
& m1_pboole(B,u1_struct_0(A))
& m4_pboole(C,u1_struct_0(A),k1_mboolean(u1_struct_0(A),B)) )
=> ( v5_closure1(g1_closure1(A,B,C),A)
& l1_closure1(g1_closure1(A,B,C),A) ) ) )).

fof(free_g1_closure1,axiom,(
! [A,B,C] :
( ( l1_struct_0(A)
& m1_pboole(B,u1_struct_0(A))
& m4_pboole(C,u1_struct_0(A),k1_mboolean(u1_struct_0(A),B)) )
=> ! [D,E,F] :
( g1_closure1(A,B,C) = g1_closure1(D,E,F)
=> ( A = D
& B = E
& C = F ) ) ) )).

fof(t1_closure1,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B,C] :
( r1_tarski(B,C)
=> r1_tarski(a_2_0_closure1(A,C),a_2_0_closure1(A,B)) ) ) )).

fof(t32_closure1,axiom,(
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( ( v5_mssubfam(C,A,B)
& m4_pboole(C,A,k1_mboolean(A,B)) )
=> ! [D] :
( m2_pboole(D,A,k7_mssubfam(A,B))
=> ! [E] :
( ( v2_relat_1(E)
& m4_pboole(E,A,k1_mboolean(A,B)) )
=> ( ! [F] :
( m1_pboole(F,A)
=> ( r1_pboole(A,F,E)
<=> ( r1_pboole(A,F,C)
& r2_pboole(A,D,F) ) ) )
=> ! [F,G] :
( ~ v1_xboole_0(G)
=> ( ( r2_hidden(F,A)
& G = k1_funct_1(C,F) )
=> k1_funct_1(E,F) = a_5_0_closure1(A,B,D,F,G) ) ) ) ) ) ) ) )).

fof(fraenkel_a_2_0_closure1,axiom,(
! [A,B,C] :
( ~ v1_xboole_0(B)
=> ( r2_hidden(A,a_2_0_closure1(B,C))
<=> ? [D] :
( m1_subset_1(D,B)
& A = D
& r1_tarski(C,D) ) ) ) )).

fof(fraenkel_a_5_0_closure1,axiom,(
! [A,B,C,D,E,F] :
( ( m1_pboole(C,B)
& m2_pboole(D,B,k7_mssubfam(B,C))
& ~ v1_xboole_0(F) )
=> ( r2_hidden(A,a_5_0_closure1(B,C,D,E,F))
<=> ? [G] :
( m1_subset_1(G,F)
& A = G
& r1_tarski(k1_funct_1(D,E),G) ) ) ) )).
%------------------------------------------------------------------------------
```