SET007 Axioms: SET007+433.ax

```%------------------------------------------------------------------------------
% File     : SET007+433 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : On the Lattice of Subgroups of a Group
% 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    : latsubgr [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   44 (   5 unit)
%            Number of atoms       :  437 (  38 equality)
%            Maximal formula depth :   18 (   9 average)
%            Number of connectives :  450 (  57 ~  ;   1  |; 236  &)
%                                         (   7 <=>; 149 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   26 (   1 propositional; 0-3 arity)
%            Number of functors    :   25 (   1 constant; 0-4 arity)
%            Number of variables   :  136 (   0 singleton; 133 !;   3 ?)
%            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(t1_latsubgr,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_group_2(B,A)
=> ! [C] :
( m1_group_2(C,A)
=> u1_struct_0(k9_group_2(A,B,C)) = k3_xboole_0(u1_struct_0(B),u1_struct_0(C)) ) ) ) )).

fof(t2_latsubgr,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( r2_hidden(B,k1_group_3(A))
<=> ? [C] :
( v1_group_1(C)
& m1_group_2(C,A)
& B = C ) ) ) )).

fof(t3_latsubgr,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( ( v1_group_1(C)
& m1_group_2(C,A) )
=> ( B = u1_struct_0(C)
=> r1_group_2(A,k5_group_4(A,B),C) ) ) ) ) )).

fof(t4_latsubgr,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_group_2(B,A)
=> ! [C] :
( m1_group_2(C,A)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( D = k2_xboole_0(u1_struct_0(B),u1_struct_0(C))
=> r1_group_2(A,k8_group_4(A,B,C),k5_group_4(A,D)) ) ) ) ) ) )).

fof(t5_latsubgr,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_group_2(B,A)
=> ! [C] :
( m1_group_2(C,A)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( ( r1_rlvect_1(B,D)
| r1_rlvect_1(C,D) )
=> r1_rlvect_1(k8_group_4(A,B,C),D) ) ) ) ) ) )).

fof(t6_latsubgr,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v1_group_6(C,A,B)
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ! [D] :
( m1_group_2(D,A)
=> ? [E] :
( v1_group_1(E)
& m1_group_2(E,B)
& u1_struct_0(E) = k2_funct_2(u1_struct_0(A),u1_struct_0(B),C,u1_struct_0(D)) ) ) ) ) ) )).

fof(t7_latsubgr,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v1_group_6(C,A,B)
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ! [D] :
( m1_group_2(D,B)
=> ? [E] :
( v1_group_1(E)
& m1_group_2(E,A)
& u1_struct_0(E) = k3_funct_2(u1_struct_0(A),u1_struct_0(B),C,u1_struct_0(D)) ) ) ) ) ) )).

fof(t8_latsubgr,axiom,(
\$true )).

fof(t9_latsubgr,axiom,(
\$true )).

fof(t10_latsubgr,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v1_group_6(C,A,B)
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ! [D] :
( m1_group_2(D,A)
=> ! [E] :
( m1_group_2(E,A)
=> ! [F] :
( m1_group_2(F,B)
=> ! [G] :
( m1_group_2(G,B)
=> ( ( u1_struct_0(F) = k2_funct_2(u1_struct_0(A),u1_struct_0(B),C,u1_struct_0(D))
& u1_struct_0(G) = k2_funct_2(u1_struct_0(A),u1_struct_0(B),C,u1_struct_0(E))
& m1_group_6(D,A,E) )
=> m1_group_6(F,B,G) ) ) ) ) ) ) ) ) )).

fof(t11_latsubgr,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v1_group_6(C,A,B)
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ! [D] :
( m1_group_2(D,B)
=> ! [E] :
( m1_group_2(E,B)
=> ! [F] :
( m1_group_2(F,A)
=> ! [G] :
( m1_group_2(G,A)
=> ( ( u1_struct_0(F) = k3_funct_2(u1_struct_0(A),u1_struct_0(B),C,u1_struct_0(D))
& u1_struct_0(G) = k3_funct_2(u1_struct_0(A),u1_struct_0(B),C,u1_struct_0(E))
& m1_group_6(D,B,E) )
=> m1_group_6(F,A,G) ) ) ) ) ) ) ) ) )).

fof(t12_latsubgr,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> r1_tarski(k2_funct_2(u1_struct_0(A),u1_struct_0(B),C,D),k2_funct_2(u1_struct_0(A),u1_struct_0(B),C,u1_struct_0(k5_group_4(A,D)))) ) ) ) ) )).

fof(t13_latsubgr,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( m1_group_2(C,A)
=> ! [D] :
( m1_group_2(D,A)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(E,u1_struct_0(A),u1_struct_0(B)) )
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A)))
=> ( F = k2_xboole_0(u1_struct_0(C),u1_struct_0(D))
=> k2_funct_2(u1_struct_0(A),u1_struct_0(B),E,u1_struct_0(k8_group_4(A,C,D))) = k2_funct_2(u1_struct_0(A),u1_struct_0(B),E,u1_struct_0(k5_group_4(A,F))) ) ) ) ) ) ) ) )).

fof(t14_latsubgr,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( B = k18_group_2(u1_struct_0(A),k2_group_1(A))
=> r1_group_2(A,k5_group_4(A,B),k5_group_2(A)) ) ) ) )).

fof(d1_latsubgr,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k1_group_3(A),k1_zfmisc_1(u1_struct_0(A)))
& m2_relset_1(B,k1_group_3(A),k1_zfmisc_1(u1_struct_0(A))) )
=> ( B = k1_latsubgr(A)
<=> ! [C] :
( ( v1_group_1(C)
& m1_group_2(C,A) )
=> k1_funct_1(B,C) = u1_struct_0(C) ) ) ) ) )).

fof(t15_latsubgr,axiom,(
\$true )).

fof(t16_latsubgr,axiom,(
\$true )).

fof(t17_latsubgr,axiom,(
\$true )).

fof(t18_latsubgr,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( v1_group_1(B)
& m1_group_2(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r2_hidden(C,k1_funct_1(k1_latsubgr(A),B))
<=> r1_rlvect_1(B,C) ) ) ) ) )).

fof(t19_latsubgr,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( v1_group_1(B)
& m1_group_2(B,A) )
=> r2_hidden(k2_group_1(A),k1_funct_1(k1_latsubgr(A),B)) ) ) )).

fof(t20_latsubgr,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( v1_group_1(B)
& m1_group_2(B,A) )
=> k1_funct_1(k1_latsubgr(A),B) != k1_xboole_0 ) ) )).

fof(t21_latsubgr,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( v1_group_1(B)
& m1_group_2(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( ( r2_hidden(C,k1_funct_1(k1_latsubgr(A),B))
& r2_hidden(D,k1_funct_1(k1_latsubgr(A),B)) )
=> r2_hidden(k1_group_1(A,C,D),k1_funct_1(k1_latsubgr(A),B)) ) ) ) ) ) )).

fof(t22_latsubgr,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( v1_group_1(B)
& m1_group_2(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r2_hidden(C,k1_funct_1(k1_latsubgr(A),B))
=> r2_hidden(k3_group_1(A,C),k1_funct_1(k1_latsubgr(A),B)) ) ) ) ) )).

fof(t23_latsubgr,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( v1_group_1(B)
& m1_group_2(B,A) )
=> ! [C] :
( ( v1_group_1(C)
& m1_group_2(C,A) )
=> u1_struct_0(k9_group_2(A,B,C)) = k3_xboole_0(k1_funct_1(k1_latsubgr(A),B),k1_funct_1(k1_latsubgr(A),C)) ) ) ) )).

fof(t24_latsubgr,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( v1_group_1(B)
& m1_group_2(B,A) )
=> ! [C] :
( ( v1_group_1(C)
& m1_group_2(C,A) )
=> k1_funct_1(k1_latsubgr(A),k9_group_2(A,B,C)) = k3_xboole_0(k1_funct_1(k1_latsubgr(A),B),k1_funct_1(k1_latsubgr(A),C)) ) ) ) )).

fof(d2_latsubgr,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_group_3(A))) )
=> ! [C] :
( ( v1_group_1(C)
& m1_group_2(C,A) )
=> ( C = k2_latsubgr(A,B)
<=> u1_struct_0(C) = k6_setfam_1(u1_struct_0(A),k2_funct_2(k1_group_3(A),k1_zfmisc_1(u1_struct_0(A)),k1_latsubgr(A),B)) ) ) ) ) )).

fof(t25_latsubgr,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_group_3(A))) )
=> ( r2_hidden(k5_group_2(A),B)
=> r1_group_2(A,k2_latsubgr(A,B),k5_group_2(A)) ) ) ) )).

fof(t26_latsubgr,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_group_3(A))
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(k1_group_3(A))) )
=> ( C = k18_group_2(k1_group_3(A),B)
=> k2_latsubgr(A,C) = B ) ) ) ) )).

fof(t27_latsubgr,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_group_2(B,A)
=> ! [C] :
( m1_group_2(C,A)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k11_group_4(A)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k11_group_4(A)))
=> ( ( D = B
& E = C )
=> k3_lattices(k11_group_4(A),D,E) = k8_group_4(A,B,C) ) ) ) ) ) ) )).

fof(t28_latsubgr,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_group_2(B,A)
=> ! [C] :
( m1_group_2(C,A)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k11_group_4(A)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k11_group_4(A)))
=> ( ( D = B
& E = C )
=> k4_lattices(k11_group_4(A),D,E) = k9_group_2(A,B,C) ) ) ) ) ) ) )).

fof(t29_latsubgr,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k11_group_4(A)))
=> ! [C] :
( m1_group_2(C,A)
=> ( B = C
=> ( v1_group_1(C)
& m1_group_2(C,A) ) ) ) ) ) )).

fof(t30_latsubgr,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_group_2(B,A)
=> ! [C] :
( m1_group_2(C,A)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k11_group_4(A)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k11_group_4(A)))
=> ( ( D = B
& E = C )
=> ( r3_lattices(k11_group_4(A),D,E)
<=> r1_tarski(u1_struct_0(B),u1_struct_0(C)) ) ) ) ) ) ) ) )).

fof(t31_latsubgr,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_group_2(B,A)
=> ! [C] :
( m1_group_2(C,A)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k11_group_4(A)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k11_group_4(A)))
=> ( ( D = B
& E = C )
=> ( r3_lattices(k11_group_4(A),D,E)
<=> m1_group_6(B,A,C) ) ) ) ) ) ) ) )).

fof(t32_latsubgr,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> v4_lattice3(k11_group_4(A)) ) )).

fof(d3_latsubgr,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(k11_group_4(A)),u1_struct_0(k11_group_4(B)))
& m2_relset_1(D,u1_struct_0(k11_group_4(A)),u1_struct_0(k11_group_4(B))) )
=> ( D = k3_latsubgr(A,B,C)
<=> ! [E] :
( ( v1_group_1(E)
& m1_group_2(E,A) )
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(B)))
=> ( F = k2_funct_2(u1_struct_0(A),u1_struct_0(B),C,u1_struct_0(E))
=> k1_funct_1(D,E) = k5_group_4(B,F) ) ) ) ) ) ) ) ) )).

fof(t33_latsubgr,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> k3_latsubgr(A,A,k6_partfun1(u1_struct_0(A))) = k6_partfun1(u1_struct_0(k11_group_4(A))) ) )).

fof(t34_latsubgr,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v1_group_6(C,A,B)
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v2_funct_1(C)
=> v2_funct_1(k3_latsubgr(A,B,C)) ) ) ) ) )).

fof(t35_latsubgr,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v1_group_6(C,A,B)
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> k1_funct_1(k3_latsubgr(A,B,C),k5_group_2(A)) = k5_group_2(B) ) ) ) )).

fof(t36_latsubgr,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v1_group_6(C,A,B)
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v2_funct_1(C)
=> m3_vectsp_8(k3_latsubgr(A,B,C),k11_group_4(A),k11_group_4(B)) ) ) ) ) )).

fof(t37_latsubgr,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v1_group_6(C,A,B)
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> m4_vectsp_8(k3_latsubgr(A,B,C),k11_group_4(A),k11_group_4(B)) ) ) ) )).

fof(t38_latsubgr,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v1_group_6(C,A,B)
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v2_funct_1(C)
=> m1_lattice4(k3_latsubgr(A,B,C),k11_group_4(A),k11_group_4(B)) ) ) ) ) )).

fof(dt_k1_latsubgr,axiom,(
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ( v1_funct_1(k1_latsubgr(A))
& v1_funct_2(k1_latsubgr(A),k1_group_3(A),k1_zfmisc_1(u1_struct_0(A)))
& m2_relset_1(k1_latsubgr(A),k1_group_3(A),k1_zfmisc_1(u1_struct_0(A))) ) ) )).

fof(dt_k2_latsubgr,axiom,(
! [A,B] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_group_3(A))) )
=> ( v1_group_1(k2_latsubgr(A,B))
& m1_group_2(k2_latsubgr(A,B),A) ) ) )).

fof(dt_k3_latsubgr,axiom,(
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A)
& ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B)
& v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m1_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v1_funct_1(k3_latsubgr(A,B,C))
& v1_funct_2(k3_latsubgr(A,B,C),u1_struct_0(k11_group_4(A)),u1_struct_0(k11_group_4(B)))
& m2_relset_1(k3_latsubgr(A,B,C),u1_struct_0(k11_group_4(A)),u1_struct_0(k11_group_4(B))) ) ) )).
%------------------------------------------------------------------------------
```