SET007 Axioms: SET007+263.ax


%------------------------------------------------------------------------------
% File     : SET007+263 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : From Double Loops to Fields
% 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    : algstr_2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   46 (  28 unit)
%            Number of atoms       :  461 (  96 equality)
%            Maximal formula depth :   26 (   6 average)
%            Number of connectives :  474 (  59 ~  ;   6  |; 245  &)
%                                         (   6 <=>; 158 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   30 (   1 propositional; 0-2 arity)
%            Number of functors    :    8 (   1 constant; 0-3 arity)
%            Number of variables   :  161 (   0 singleton; 155 !;   6 ?)
%            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(fc1_algstr_2,axiom,
    ( ~ v3_struct_0(k3_vectsp_1)
    & v3_rlvect_1(k3_vectsp_1)
    & v4_rlvect_1(k3_vectsp_1)
    & v5_rlvect_1(k3_vectsp_1)
    & v6_rlvect_1(k3_vectsp_1)
    & v2_group_1(k3_vectsp_1)
    & v4_group_1(k3_vectsp_1)
    & v7_group_1(k3_vectsp_1)
    & v3_vectsp_1(k3_vectsp_1)
    & v4_vectsp_1(k3_vectsp_1)
    & v5_vectsp_1(k3_vectsp_1)
    & v6_vectsp_1(k3_vectsp_1)
    & v7_vectsp_1(k3_vectsp_1)
    & v8_vectsp_1(k3_vectsp_1)
    & v9_vectsp_1(k3_vectsp_1)
    & ~ v10_vectsp_1(k3_vectsp_1)
    & v1_algstr_1(k3_vectsp_1)
    & v2_algstr_1(k3_vectsp_1)
    & v3_algstr_1(k3_vectsp_1)
    & v4_algstr_1(k3_vectsp_1)
    & v5_algstr_1(k3_vectsp_1)
    & v6_algstr_1(k3_vectsp_1)
    & v9_algstr_1(k3_vectsp_1)
    & v10_algstr_1(k3_vectsp_1)
    & v11_algstr_1(k3_vectsp_1) )).

fof(rc1_algstr_2,axiom,(
    ? [A] :
      ( l3_vectsp_1(A)
      & ~ v3_struct_0(A)
      & v3_rlvect_1(A)
      & v4_rlvect_1(A)
      & v5_rlvect_1(A)
      & v6_rlvect_1(A)
      & v2_group_1(A)
      & v4_group_1(A)
      & v7_group_1(A)
      & v3_vectsp_1(A)
      & v4_vectsp_1(A)
      & v5_vectsp_1(A)
      & v6_vectsp_1(A)
      & v7_vectsp_1(A)
      & v8_vectsp_1(A)
      & ~ v10_vectsp_1(A)
      & v1_algstr_1(A)
      & v2_algstr_1(A)
      & v3_algstr_1(A)
      & v4_algstr_1(A)
      & v5_algstr_1(A)
      & v6_algstr_1(A)
      & v9_algstr_1(A)
      & v10_algstr_1(A)
      & v11_algstr_1(A) ) )).

fof(d1_algstr_2,axiom,(
    $true )).

fof(d2_algstr_2,axiom,(
    $true )).

fof(d3_algstr_2,axiom,(
    $true )).

fof(d4_algstr_2,axiom,(
    $true )).

fof(d5_algstr_2,axiom,(
    $true )).

fof(d6_algstr_2,axiom,(
    $true )).

fof(d7_algstr_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_algstr_1(A)
        & v5_algstr_1(A)
        & l1_rlvect_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( C = k1_algstr_2(A,B)
              <=> k2_rlvect_1(A,B,C) = k1_rlvect_1(A) ) ) ) ) )).

fof(d8_algstr_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_algstr_1(A)
        & v5_algstr_1(A)
        & l1_rlvect_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => k2_algstr_2(A,B,C) = k2_rlvect_1(A,B,k1_algstr_2(A,C)) ) ) ) )).

fof(t1_algstr_2,axiom,(
    $true )).

fof(t2_algstr_2,axiom,(
    $true )).

fof(t3_algstr_2,axiom,(
    $true )).

fof(t4_algstr_2,axiom,(
    $true )).

fof(t5_algstr_2,axiom,(
    $true )).

fof(t6_algstr_2,axiom,(
    $true )).

fof(t7_algstr_2,axiom,(
    $true )).

fof(t8_algstr_2,axiom,(
    $true )).

fof(t9_algstr_2,axiom,(
    $true )).

fof(t10_algstr_2,axiom,(
    $true )).

fof(t11_algstr_2,axiom,(
    $true )).

fof(t12_algstr_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ( ( ~ v3_struct_0(A)
          & v3_rlvect_1(A)
          & v4_rlvect_1(A)
          & v5_rlvect_1(A)
          & v2_group_1(A)
          & v4_vectsp_1(A)
          & ~ v10_vectsp_1(A)
          & v1_algstr_1(A)
          & v6_algstr_1(A)
          & v11_algstr_1(A)
          & l3_vectsp_1(A) )
      <=> ( ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => k2_rlvect_1(A,B,k1_rlvect_1(A)) = B )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => ? [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                  & k2_rlvect_1(A,B,C) = k1_rlvect_1(A) ) )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ! [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                     => k2_rlvect_1(A,k2_rlvect_1(A,B,C),D) = k2_rlvect_1(A,B,k2_rlvect_1(A,C,D)) ) ) )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => k2_rlvect_1(A,B,C) = k2_rlvect_1(A,C,B) ) )
          & k1_rlvect_1(A) != k2_group_1(A)
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => k1_group_1(A,B,k2_group_1(A)) = B )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => k1_group_1(A,k2_group_1(A),B) = B )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ~ ( B != k1_rlvect_1(A)
                      & ! [D] :
                          ( m1_subset_1(D,u1_struct_0(A))
                         => k1_group_1(A,B,D) != C ) ) ) )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ~ ( B != k1_rlvect_1(A)
                      & ! [D] :
                          ( m1_subset_1(D,u1_struct_0(A))
                         => k1_group_1(A,D,B) != C ) ) ) )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ! [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                     => ( k1_group_1(A,B,C) = k1_group_1(A,B,D)
                       => ( B = k1_rlvect_1(A)
                          | C = D ) ) ) ) )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ! [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                     => ( k1_group_1(A,C,B) = k1_group_1(A,D,B)
                       => ( B = k1_rlvect_1(A)
                          | C = D ) ) ) ) )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => k1_group_1(A,B,k1_rlvect_1(A)) = k1_rlvect_1(A) )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => k1_group_1(A,k1_rlvect_1(A),B) = k1_rlvect_1(A) )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ! [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                     => k1_group_1(A,B,k2_rlvect_1(A,C,D)) = k2_rlvect_1(A,k1_group_1(A,B,C),k1_group_1(A,B,D)) ) ) ) ) ) ) )).

fof(t13_algstr_2,axiom,(
    $true )).

fof(t14_algstr_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v5_rlvect_1(A)
        & v2_group_1(A)
        & v4_vectsp_1(A)
        & v1_algstr_1(A)
        & v6_algstr_1(A)
        & v11_algstr_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => k1_group_1(A,B,k1_algstr_2(A,C)) = k1_algstr_2(A,k1_group_1(A,B,C)) ) ) ) )).

fof(t15_algstr_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v2_algstr_1(A)
        & v5_algstr_1(A)
        & l1_rlvect_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => k1_algstr_2(A,k1_algstr_2(A,B)) = B ) ) )).

fof(t16_algstr_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v5_rlvect_1(A)
        & v2_group_1(A)
        & v4_vectsp_1(A)
        & v1_algstr_1(A)
        & v6_algstr_1(A)
        & v11_algstr_1(A)
        & l3_vectsp_1(A) )
     => k1_group_1(A,k1_algstr_2(A,k2_group_1(A)),k1_algstr_2(A,k2_group_1(A))) = k2_group_1(A) ) )).

fof(t17_algstr_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v5_rlvect_1(A)
        & v2_group_1(A)
        & v4_vectsp_1(A)
        & v1_algstr_1(A)
        & v6_algstr_1(A)
        & v11_algstr_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => k1_group_1(A,B,k2_algstr_2(A,C,D)) = k2_algstr_2(A,k1_group_1(A,B,C),k1_group_1(A,B,D)) ) ) ) ) )).

fof(t18_algstr_2,axiom,(
    $true )).

fof(t19_algstr_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ( ( ~ v3_struct_0(A)
          & v3_rlvect_1(A)
          & v4_rlvect_1(A)
          & v5_rlvect_1(A)
          & v2_group_1(A)
          & v5_vectsp_1(A)
          & ~ v10_vectsp_1(A)
          & v1_algstr_1(A)
          & v6_algstr_1(A)
          & v11_algstr_1(A)
          & l3_vectsp_1(A) )
      <=> ( ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => k2_rlvect_1(A,B,k1_rlvect_1(A)) = B )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => ? [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                  & k2_rlvect_1(A,B,C) = k1_rlvect_1(A) ) )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ! [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                     => k2_rlvect_1(A,k2_rlvect_1(A,B,C),D) = k2_rlvect_1(A,B,k2_rlvect_1(A,C,D)) ) ) )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => k2_rlvect_1(A,B,C) = k2_rlvect_1(A,C,B) ) )
          & k1_rlvect_1(A) != k2_group_1(A)
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => k1_group_1(A,B,k2_group_1(A)) = B )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => k1_group_1(A,k2_group_1(A),B) = B )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ~ ( B != k1_rlvect_1(A)
                      & ! [D] :
                          ( m1_subset_1(D,u1_struct_0(A))
                         => k1_group_1(A,B,D) != C ) ) ) )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ~ ( B != k1_rlvect_1(A)
                      & ! [D] :
                          ( m1_subset_1(D,u1_struct_0(A))
                         => k1_group_1(A,D,B) != C ) ) ) )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ! [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                     => ( k1_group_1(A,B,C) = k1_group_1(A,B,D)
                       => ( B = k1_rlvect_1(A)
                          | C = D ) ) ) ) )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ! [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                     => ( k1_group_1(A,C,B) = k1_group_1(A,D,B)
                       => ( B = k1_rlvect_1(A)
                          | C = D ) ) ) ) )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => k1_group_1(A,B,k1_rlvect_1(A)) = k1_rlvect_1(A) )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => k1_group_1(A,k1_rlvect_1(A),B) = k1_rlvect_1(A) )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ! [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                     => k1_group_1(A,k2_rlvect_1(A,C,D),B) = k2_rlvect_1(A,k1_group_1(A,C,B),k1_group_1(A,D,B)) ) ) ) ) ) ) )).

fof(t20_algstr_2,axiom,(
    $true )).

fof(t21_algstr_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v5_rlvect_1(A)
        & v2_group_1(A)
        & v5_vectsp_1(A)
        & v1_algstr_1(A)
        & v6_algstr_1(A)
        & v11_algstr_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => k1_group_1(A,k1_algstr_2(A,B),C) = k1_algstr_2(A,k1_group_1(A,B,C)) ) ) ) )).

fof(t22_algstr_2,axiom,(
    $true )).

fof(t23_algstr_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v5_rlvect_1(A)
        & v2_group_1(A)
        & v5_vectsp_1(A)
        & v1_algstr_1(A)
        & v6_algstr_1(A)
        & v11_algstr_1(A)
        & l3_vectsp_1(A) )
     => k1_group_1(A,k1_algstr_2(A,k2_group_1(A)),k1_algstr_2(A,k2_group_1(A))) = k2_group_1(A) ) )).

fof(t24_algstr_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v5_rlvect_1(A)
        & v2_group_1(A)
        & v5_vectsp_1(A)
        & v1_algstr_1(A)
        & v6_algstr_1(A)
        & v11_algstr_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => k1_group_1(A,k2_algstr_2(A,B,C),D) = k2_algstr_2(A,k1_group_1(A,B,D),k1_group_1(A,C,D)) ) ) ) ) )).

fof(t25_algstr_2,axiom,(
    $true )).

fof(t26_algstr_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ( ( ~ v3_struct_0(A)
          & v3_rlvect_1(A)
          & v4_rlvect_1(A)
          & v5_rlvect_1(A)
          & v2_group_1(A)
          & v7_vectsp_1(A)
          & ~ v10_vectsp_1(A)
          & v1_algstr_1(A)
          & v6_algstr_1(A)
          & v11_algstr_1(A)
          & l3_vectsp_1(A) )
      <=> ( ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => k2_rlvect_1(A,B,k1_rlvect_1(A)) = B )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => ? [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                  & k2_rlvect_1(A,B,C) = k1_rlvect_1(A) ) )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ! [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                     => k2_rlvect_1(A,k2_rlvect_1(A,B,C),D) = k2_rlvect_1(A,B,k2_rlvect_1(A,C,D)) ) ) )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => k2_rlvect_1(A,B,C) = k2_rlvect_1(A,C,B) ) )
          & k1_rlvect_1(A) != k2_group_1(A)
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => k1_group_1(A,B,k2_group_1(A)) = B )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => k1_group_1(A,k2_group_1(A),B) = B )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ~ ( B != k1_rlvect_1(A)
                      & ! [D] :
                          ( m1_subset_1(D,u1_struct_0(A))
                         => k1_group_1(A,B,D) != C ) ) ) )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ~ ( B != k1_rlvect_1(A)
                      & ! [D] :
                          ( m1_subset_1(D,u1_struct_0(A))
                         => k1_group_1(A,D,B) != C ) ) ) )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ! [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                     => ( k1_group_1(A,B,C) = k1_group_1(A,B,D)
                       => ( B = k1_rlvect_1(A)
                          | C = D ) ) ) ) )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ! [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                     => ( k1_group_1(A,C,B) = k1_group_1(A,D,B)
                       => ( B = k1_rlvect_1(A)
                          | C = D ) ) ) ) )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => k1_group_1(A,B,k1_rlvect_1(A)) = k1_rlvect_1(A) )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => k1_group_1(A,k1_rlvect_1(A),B) = k1_rlvect_1(A) )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ! [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                     => k1_group_1(A,B,k2_rlvect_1(A,C,D)) = k2_rlvect_1(A,k1_group_1(A,B,C),k1_group_1(A,B,D)) ) ) )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ! [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                     => k1_group_1(A,k2_rlvect_1(A,C,D),B) = k2_rlvect_1(A,k1_group_1(A,C,B),k1_group_1(A,D,B)) ) ) ) ) ) ) )).

fof(t27_algstr_2,axiom,(
    $true )).

fof(t28_algstr_2,axiom,(
    $true )).

fof(t29_algstr_2,axiom,(
    $true )).

fof(t30_algstr_2,axiom,(
    $true )).

fof(t31_algstr_2,axiom,(
    $true )).

fof(t32_algstr_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ( ( ~ v3_struct_0(A)
          & v3_rlvect_1(A)
          & v4_rlvect_1(A)
          & v5_rlvect_1(A)
          & v2_group_1(A)
          & v4_group_1(A)
          & v7_vectsp_1(A)
          & ~ v10_vectsp_1(A)
          & v1_algstr_1(A)
          & v6_algstr_1(A)
          & v11_algstr_1(A)
          & l3_vectsp_1(A) )
      <=> ( ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => k2_rlvect_1(A,B,k1_rlvect_1(A)) = B )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => ? [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                  & k2_rlvect_1(A,B,C) = k1_rlvect_1(A) ) )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ! [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                     => k2_rlvect_1(A,k2_rlvect_1(A,B,C),D) = k2_rlvect_1(A,B,k2_rlvect_1(A,C,D)) ) ) )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => k2_rlvect_1(A,B,C) = k2_rlvect_1(A,C,B) ) )
          & k1_rlvect_1(A) != k2_group_1(A)
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => k1_group_1(A,B,k2_group_1(A)) = B )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => ~ ( B != k1_rlvect_1(A)
                  & ! [C] :
                      ( m1_subset_1(C,u1_struct_0(A))
                     => k1_group_1(A,B,C) != k2_group_1(A) ) ) )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => k1_group_1(A,B,k1_rlvect_1(A)) = k1_rlvect_1(A) )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => k1_group_1(A,k1_rlvect_1(A),B) = k1_rlvect_1(A) )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ! [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                     => k1_group_1(A,k1_group_1(A,B,C),D) = k1_group_1(A,B,k1_group_1(A,C,D)) ) ) )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ! [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                     => k1_group_1(A,B,k2_rlvect_1(A,C,D)) = k2_rlvect_1(A,k1_group_1(A,B,C),k1_group_1(A,B,D)) ) ) )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ! [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                     => k1_group_1(A,k2_rlvect_1(A,C,D),B) = k2_rlvect_1(A,k1_group_1(A,C,B),k1_group_1(A,D,B)) ) ) ) ) ) ) )).

fof(t33_algstr_2,axiom,(
    $true )).

fof(t34_algstr_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ( ( ~ v3_struct_0(A)
          & v3_rlvect_1(A)
          & v4_rlvect_1(A)
          & v5_rlvect_1(A)
          & v2_group_1(A)
          & v4_group_1(A)
          & v7_group_1(A)
          & v7_vectsp_1(A)
          & ~ v10_vectsp_1(A)
          & v1_algstr_1(A)
          & v6_algstr_1(A)
          & v11_algstr_1(A)
          & l3_vectsp_1(A) )
      <=> ( ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => k2_rlvect_1(A,B,k1_rlvect_1(A)) = B )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => ? [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                  & k2_rlvect_1(A,B,C) = k1_rlvect_1(A) ) )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ! [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                     => k2_rlvect_1(A,k2_rlvect_1(A,B,C),D) = k2_rlvect_1(A,B,k2_rlvect_1(A,C,D)) ) ) )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => k2_rlvect_1(A,B,C) = k2_rlvect_1(A,C,B) ) )
          & k1_rlvect_1(A) != k2_group_1(A)
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => k1_group_1(A,B,k2_group_1(A)) = B )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => ~ ( B != k1_rlvect_1(A)
                  & ! [C] :
                      ( m1_subset_1(C,u1_struct_0(A))
                     => k1_group_1(A,B,C) != k2_group_1(A) ) ) )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => k1_group_1(A,B,k1_rlvect_1(A)) = k1_rlvect_1(A) )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ! [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                     => k1_group_1(A,k1_group_1(A,B,C),D) = k1_group_1(A,B,k1_group_1(A,C,D)) ) ) )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ! [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                     => k1_group_1(A,B,k2_rlvect_1(A,C,D)) = k2_rlvect_1(A,k1_group_1(A,B,C),k1_group_1(A,B,D)) ) ) )
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => k1_group_1(A,B,C) = k1_group_1(A,C,B) ) ) ) ) ) )).

fof(dt_k1_algstr_2,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v2_algstr_1(A)
        & v5_algstr_1(A)
        & l1_rlvect_1(A)
        & m1_subset_1(B,u1_struct_0(A)) )
     => m1_subset_1(k1_algstr_2(A,B),u1_struct_0(A)) ) )).

fof(dt_k2_algstr_2,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v2_algstr_1(A)
        & v5_algstr_1(A)
        & l1_rlvect_1(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => m1_subset_1(k2_algstr_2(A,B,C),u1_struct_0(A)) ) )).
%------------------------------------------------------------------------------