SET007 Axioms: SET007+394.ax


%------------------------------------------------------------------------------
% File     : SET007+394 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : On the Group of Inner Automorphisms
% 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    : autgroup [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   44 (   2 unit)
%            Number of atoms       :  353 (  34 equality)
%            Maximal formula depth :   13 (   8 average)
%            Number of connectives :  353 (  44 ~  ;   0  |; 204  &)
%                                         (   9 <=>;  96 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   24 (   1 propositional; 0-4 arity)
%            Number of functors    :   22 (   0 constant; 1-6 arity)
%            Number of variables   :   92 (   0 singleton;  91 !;   1 ?)
%            Maximal term depth    :    4 (   2 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_autgroup,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_group_2(B,A)
         => ( ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ( m1_subset_1(D,u1_struct_0(B))
                     => r1_rlvect_1(B,k2_group_3(A,D,C)) ) ) )
          <=> v1_group_3(B,A) ) ) ) )).

fof(d1_autgroup,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_fraenkel(B,u1_struct_0(A),u1_struct_0(A))
         => ( B = k1_autgroup(A)
          <=> ( ! [C] :
                  ( m2_fraenkel(C,u1_struct_0(A),u1_struct_0(A),B)
                 => ( v1_funct_1(C)
                    & v1_funct_2(C,u1_struct_0(A),u1_struct_0(A))
                    & v1_group_6(C,A,A)
                    & m2_relset_1(C,u1_struct_0(A),u1_struct_0(A)) ) )
              & ! [C] :
                  ( ( v1_funct_1(C)
                    & v1_funct_2(C,u1_struct_0(A),u1_struct_0(A))
                    & v1_group_6(C,A,A)
                    & m2_relset_1(C,u1_struct_0(A),u1_struct_0(A)) )
                 => ( r2_hidden(C,B)
                  <=> ( v2_funct_1(C)
                      & v3_group_6(C,A,A) ) ) ) ) ) ) ) )).

fof(t2_autgroup,axiom,(
    $true )).

fof(t3_autgroup,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => r1_tarski(k1_autgroup(A),k1_fraenkel(u1_struct_0(A),u1_struct_0(A))) ) )).

fof(t4_autgroup,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => m2_fraenkel(k6_partfun1(u1_struct_0(A)),u1_struct_0(A),u1_struct_0(A),k1_autgroup(A)) ) )).

fof(t5_autgroup,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
            & v1_group_6(B,A,A)
            & m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
         => ( r2_hidden(B,k1_autgroup(A))
          <=> v4_group_6(B,A,A) ) ) ) )).

fof(t6_autgroup,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m2_fraenkel(B,u1_struct_0(A),u1_struct_0(A),k1_autgroup(A))
         => ( v1_funct_1(k2_funct_1(B))
            & v1_funct_2(k2_funct_1(B),u1_struct_0(A),u1_struct_0(A))
            & v1_group_6(k2_funct_1(B),A,A)
            & m2_relset_1(k2_funct_1(B),u1_struct_0(A),u1_struct_0(A)) ) ) ) )).

fof(t7_autgroup,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m2_fraenkel(B,u1_struct_0(A),u1_struct_0(A),k1_autgroup(A))
         => m2_fraenkel(k2_funct_1(B),u1_struct_0(A),u1_struct_0(A),k1_autgroup(A)) ) ) )).

fof(t8_autgroup,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m2_fraenkel(B,u1_struct_0(A),u1_struct_0(A),k1_autgroup(A))
         => ! [C] :
              ( m2_fraenkel(C,u1_struct_0(A),u1_struct_0(A),k1_autgroup(A))
             => m2_fraenkel(k7_funct_2(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),C,B),u1_struct_0(A),u1_struct_0(A),k1_autgroup(A)) ) ) ) )).

fof(d2_autgroup,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(k1_autgroup(A),k1_autgroup(A)),k1_autgroup(A))
            & m2_relset_1(B,k2_zfmisc_1(k1_autgroup(A),k1_autgroup(A)),k1_autgroup(A)) )
         => ( B = k2_autgroup(A)
          <=> ! [C] :
                ( m2_fraenkel(C,u1_struct_0(A),u1_struct_0(A),k1_autgroup(A))
               => ! [D] :
                    ( m2_fraenkel(D,u1_struct_0(A),u1_struct_0(A),k1_autgroup(A))
                   => k2_binop_1(k1_autgroup(A),k1_autgroup(A),k1_autgroup(A),B,C,D) = k7_funct_2(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),D,C) ) ) ) ) ) )).

fof(d3_autgroup,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => k3_autgroup(A) = g1_group_1(k1_autgroup(A),k2_autgroup(A)) ) )).

fof(t9_autgroup,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k3_autgroup(A)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k3_autgroup(A)))
             => ! [D] :
                  ( m2_fraenkel(D,u1_struct_0(A),u1_struct_0(A),k1_autgroup(A))
                 => ! [E] :
                      ( m2_fraenkel(E,u1_struct_0(A),u1_struct_0(A),k1_autgroup(A))
                     => ( ( B = D
                          & C = E )
                       => k1_group_1(k3_autgroup(A),B,C) = k7_funct_2(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),E,D) ) ) ) ) ) ) )).

fof(t10_autgroup,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => k6_partfun1(u1_struct_0(A)) = k2_group_1(k3_autgroup(A)) ) )).

fof(t11_autgroup,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m2_fraenkel(B,u1_struct_0(A),u1_struct_0(A),k1_autgroup(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k3_autgroup(A)))
             => ( B = C
               => k2_funct_1(B) = k3_group_1(k3_autgroup(A),C) ) ) ) ) )).

fof(d4_autgroup,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_fraenkel(B,u1_struct_0(A),u1_struct_0(A))
         => ( B = k4_autgroup(A)
          <=> ! [C] :
                ( m2_fraenkel(C,u1_struct_0(A),u1_struct_0(A),k1_fraenkel(u1_struct_0(A),u1_struct_0(A)))
               => ( r2_hidden(C,B)
                <=> ? [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                      & ! [E] :
                          ( m1_subset_1(E,u1_struct_0(A))
                         => k8_funct_2(u1_struct_0(A),u1_struct_0(A),C,E) = k2_group_3(A,E,D) ) ) ) ) ) ) ) )).

fof(t12_autgroup,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => r1_tarski(k4_autgroup(A),k1_fraenkel(u1_struct_0(A),u1_struct_0(A))) ) )).

fof(t13_autgroup,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m2_fraenkel(B,u1_struct_0(A),u1_struct_0(A),k4_autgroup(A))
         => m2_fraenkel(B,u1_struct_0(A),u1_struct_0(A),k1_autgroup(A)) ) ) )).

fof(t14_autgroup,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => r1_tarski(k4_autgroup(A),k1_autgroup(A)) ) )).

fof(t15_autgroup,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m2_fraenkel(B,u1_struct_0(A),u1_struct_0(A),k4_autgroup(A))
         => ! [C] :
              ( m2_fraenkel(C,u1_struct_0(A),u1_struct_0(A),k4_autgroup(A))
             => k1_binop_1(k2_autgroup(A),B,C) = k7_funct_2(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),C,B) ) ) ) )).

fof(t16_autgroup,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => m2_fraenkel(k6_partfun1(u1_struct_0(A)),u1_struct_0(A),u1_struct_0(A),k4_autgroup(A)) ) )).

fof(t17_autgroup,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m2_fraenkel(B,u1_struct_0(A),u1_struct_0(A),k4_autgroup(A))
         => m2_fraenkel(k2_funct_1(B),u1_struct_0(A),u1_struct_0(A),k4_autgroup(A)) ) ) )).

fof(t18_autgroup,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m2_fraenkel(B,u1_struct_0(A),u1_struct_0(A),k4_autgroup(A))
         => ! [C] :
              ( m2_fraenkel(C,u1_struct_0(A),u1_struct_0(A),k4_autgroup(A))
             => m2_fraenkel(k7_funct_2(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),C,B),u1_struct_0(A),u1_struct_0(A),k4_autgroup(A)) ) ) ) )).

fof(d5_autgroup,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( ( v1_group_1(B)
            & v1_group_3(B,k3_autgroup(A))
            & m1_group_2(B,k3_autgroup(A)) )
         => ( B = k5_autgroup(A)
          <=> u1_struct_0(B) = k4_autgroup(A) ) ) ) )).

fof(t19_autgroup,axiom,(
    $true )).

fof(t20_autgroup,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k5_autgroup(A)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k5_autgroup(A)))
             => ! [D] :
                  ( m2_fraenkel(D,u1_struct_0(A),u1_struct_0(A),k4_autgroup(A))
                 => ! [E] :
                      ( m2_fraenkel(E,u1_struct_0(A),u1_struct_0(A),k4_autgroup(A))
                     => ( ( B = D
                          & C = E )
                       => k1_group_1(k5_autgroup(A),B,C) = k7_funct_2(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),E,D) ) ) ) ) ) ) )).

fof(t21_autgroup,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => k6_partfun1(u1_struct_0(A)) = k2_group_1(k5_autgroup(A)) ) )).

fof(t22_autgroup,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m2_fraenkel(B,u1_struct_0(A),u1_struct_0(A),k4_autgroup(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k5_autgroup(A)))
             => ( B = C
               => k2_funct_1(B) = k3_group_1(k5_autgroup(A),C) ) ) ) ) )).

fof(d6_autgroup,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m2_fraenkel(C,u1_struct_0(A),u1_struct_0(A),k4_autgroup(A))
             => ( C = k6_autgroup(A,B)
              <=> ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => k8_funct_2(u1_struct_0(A),u1_struct_0(A),C,D) = k2_group_3(A,D,B) ) ) ) ) ) )).

fof(t23_autgroup,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => k6_autgroup(A,k1_group_1(A,B,C)) = k7_funct_2(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),k6_autgroup(A,B),k6_autgroup(A,C)) ) ) ) )).

fof(t24_autgroup,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => k6_autgroup(A,k2_group_1(A)) = k6_partfun1(u1_struct_0(A)) ) )).

fof(t25_autgroup,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => k8_funct_2(u1_struct_0(A),u1_struct_0(A),k6_autgroup(A,k2_group_1(A)),B) = B ) ) )).

fof(t26_autgroup,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => k7_funct_2(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),k6_autgroup(A,k3_group_1(A,B)),k6_autgroup(A,B)) = k6_autgroup(A,k2_group_1(A)) ) ) )).

fof(t27_autgroup,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => k7_funct_2(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),k6_autgroup(A,B),k6_autgroup(A,k3_group_1(A,B))) = k6_autgroup(A,k2_group_1(A)) ) ) )).

fof(t28_autgroup,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => k6_autgroup(A,k3_group_1(A,B)) = k2_funct_1(k6_autgroup(A,B)) ) ) )).

fof(t29_autgroup,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( k7_funct_2(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),k6_autgroup(A,k2_group_1(A)),k6_autgroup(A,B)) = k6_autgroup(A,B)
            & k7_funct_2(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),k6_autgroup(A,B),k6_autgroup(A,k2_group_1(A))) = k6_autgroup(A,B) ) ) ) )).

fof(t30_autgroup,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m2_fraenkel(B,u1_struct_0(A),u1_struct_0(A),k4_autgroup(A))
         => ( k7_funct_2(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),k6_autgroup(A,k2_group_1(A)),B) = B
            & k7_funct_2(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),B,k6_autgroup(A,k2_group_1(A))) = B ) ) ) )).

fof(t31_autgroup,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => r2_group_6(k5_autgroup(A),k6_group_6(A,k10_group_5(A))) ) )).

fof(t32_autgroup,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ( ( ~ v3_struct_0(A)
          & v3_group_1(A)
          & v4_group_1(A)
          & v7_group_1(A)
          & l1_group_1(A) )
       => ! [B] :
            ( m1_subset_1(B,u1_struct_0(k5_autgroup(A)))
           => B = k2_group_1(k5_autgroup(A)) ) ) ) )).

fof(dt_k1_autgroup,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => m1_fraenkel(k1_autgroup(A),u1_struct_0(A),u1_struct_0(A)) ) )).

fof(dt_k2_autgroup,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ( v1_funct_1(k2_autgroup(A))
        & v1_funct_2(k2_autgroup(A),k2_zfmisc_1(k1_autgroup(A),k1_autgroup(A)),k1_autgroup(A))
        & m2_relset_1(k2_autgroup(A),k2_zfmisc_1(k1_autgroup(A),k1_autgroup(A)),k1_autgroup(A)) ) ) )).

fof(dt_k3_autgroup,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ( ~ v3_struct_0(k3_autgroup(A))
        & v1_group_1(k3_autgroup(A))
        & v3_group_1(k3_autgroup(A))
        & v4_group_1(k3_autgroup(A))
        & l1_group_1(k3_autgroup(A)) ) ) )).

fof(dt_k4_autgroup,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => m1_fraenkel(k4_autgroup(A),u1_struct_0(A),u1_struct_0(A)) ) )).

fof(dt_k5_autgroup,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ( v1_group_1(k5_autgroup(A))
        & v1_group_3(k5_autgroup(A),k3_autgroup(A))
        & m1_group_2(k5_autgroup(A),k3_autgroup(A)) ) ) )).

fof(dt_k6_autgroup,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v1_group_1(A)
        & v3_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A)
        & m1_subset_1(B,u1_struct_0(A)) )
     => m2_fraenkel(k6_autgroup(A,B),u1_struct_0(A),u1_struct_0(A),k4_autgroup(A)) ) )).
%------------------------------------------------------------------------------