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))) ) ) )).
%------------------------------------------------------------------------------