SET007 Axioms: SET007+243.ax


%------------------------------------------------------------------------------
% File     : SET007+243 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : From Loops to Abelian Multiplicative Groups with Zero
% 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_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   98 (  33 unit)
%            Number of atoms       :  656 ( 109 equality)
%            Maximal formula depth :   17 (   7 average)
%            Number of connectives :  658 ( 100 ~  ;   4  |; 332  &)
%                                         (  21 <=>; 201 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   31 (   1 propositional; 0-2 arity)
%            Number of functors    :   24 (   9 constant; 0-4 arity)
%            Number of variables   :  212 (   0 singleton; 189 !;  23 ?)
%            Maximal term depth    :    3 (   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(fc1_algstr_1,axiom,
    ( ~ v3_struct_0(k2_algstr_1)
    & v1_rlvect_1(k2_algstr_1) )).

fof(cc1_algstr_1,axiom,(
    ! [A] :
      ( l1_rlvect_1(A)
     => ( ( ~ v3_struct_0(A)
          & v6_algstr_1(A) )
       => ( ~ v3_struct_0(A)
          & v2_algstr_1(A)
          & v3_algstr_1(A)
          & v4_algstr_1(A)
          & v5_algstr_1(A) ) ) ) )).

fof(cc2_algstr_1,axiom,(
    ! [A] :
      ( l1_rlvect_1(A)
     => ( ( ~ v3_struct_0(A)
          & v2_algstr_1(A)
          & v3_algstr_1(A)
          & v4_algstr_1(A)
          & v5_algstr_1(A) )
       => ( ~ v3_struct_0(A)
          & v6_algstr_1(A) ) ) ) )).

fof(fc2_algstr_1,axiom,
    ( ~ v3_struct_0(k2_algstr_1)
    & v1_rlvect_1(k2_algstr_1)
    & v4_rlvect_1(k2_algstr_1)
    & v5_rlvect_1(k2_algstr_1)
    & v1_algstr_1(k2_algstr_1)
    & v2_algstr_1(k2_algstr_1)
    & v3_algstr_1(k2_algstr_1)
    & v4_algstr_1(k2_algstr_1)
    & v5_algstr_1(k2_algstr_1)
    & v6_algstr_1(k2_algstr_1) )).

fof(rc1_algstr_1,axiom,(
    ? [A] :
      ( l1_rlvect_1(A)
      & ~ v3_struct_0(A)
      & v1_rlvect_1(A)
      & v5_rlvect_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) ) )).

fof(rc2_algstr_1,axiom,(
    ? [A] :
      ( l1_rlvect_1(A)
      & ~ v3_struct_0(A)
      & v1_rlvect_1(A)
      & v4_rlvect_1(A)
      & v5_rlvect_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) ) )).

fof(cc3_algstr_1,axiom,(
    ! [A] :
      ( l1_rlvect_1(A)
     => ( ( ~ v3_struct_0(A)
          & v6_algstr_1(A) )
       => ( ~ v3_struct_0(A)
          & v6_rlvect_1(A) ) ) ) )).

fof(cc4_algstr_1,axiom,(
    ! [A] :
      ( l1_rlvect_1(A)
     => ( ( ~ v3_struct_0(A)
          & v4_rlvect_1(A)
          & v5_rlvect_1(A)
          & v6_rlvect_1(A) )
       => ( ~ v3_struct_0(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) ) ) ) )).

fof(fc3_algstr_1,axiom,
    ( ~ v3_struct_0(k2_algstr_1)
    & v1_rlvect_1(k2_algstr_1)
    & v3_rlvect_1(k2_algstr_1)
    & v4_rlvect_1(k2_algstr_1)
    & v5_rlvect_1(k2_algstr_1)
    & v6_rlvect_1(k2_algstr_1)
    & v1_algstr_1(k2_algstr_1)
    & v2_algstr_1(k2_algstr_1)
    & v3_algstr_1(k2_algstr_1)
    & v4_algstr_1(k2_algstr_1)
    & v5_algstr_1(k2_algstr_1)
    & v6_algstr_1(k2_algstr_1) )).

fof(rc3_algstr_1,axiom,(
    ? [A] :
      ( l1_rlvect_1(A)
      & ~ v3_struct_0(A)
      & v1_rlvect_1(A)
      & v3_rlvect_1(A)
      & v4_rlvect_1(A)
      & v5_rlvect_1(A)
      & v6_rlvect_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) ) )).

fof(fc4_algstr_1,axiom,
    ( ~ v3_struct_0(k3_algstr_1)
    & v1_vectsp_1(k3_algstr_1) )).

fof(rc4_algstr_1,axiom,(
    ? [A] :
      ( l1_vectsp_1(A)
      & ~ v3_struct_0(A)
      & v2_group_1(A)
      & v1_vectsp_1(A)
      & v6_vectsp_1(A)
      & v8_vectsp_1(A)
      & v7_algstr_1(A)
      & v8_algstr_1(A) ) )).

fof(fc5_algstr_1,axiom,
    ( ~ v3_struct_0(k3_algstr_1)
    & v2_group_1(k3_algstr_1)
    & v1_vectsp_1(k3_algstr_1)
    & v6_vectsp_1(k3_algstr_1)
    & v8_vectsp_1(k3_algstr_1)
    & v7_algstr_1(k3_algstr_1)
    & v8_algstr_1(k3_algstr_1) )).

fof(rc5_algstr_1,axiom,(
    ? [A] :
      ( l1_vectsp_1(A)
      & ~ v3_struct_0(A)
      & v2_group_1(A)
      & v4_group_1(A)
      & v1_vectsp_1(A)
      & v6_vectsp_1(A)
      & v8_vectsp_1(A)
      & v7_algstr_1(A)
      & v8_algstr_1(A) ) )).

fof(fc6_algstr_1,axiom,
    ( ~ v3_struct_0(k3_algstr_1)
    & v2_group_1(k3_algstr_1)
    & v4_group_1(k3_algstr_1)
    & v1_vectsp_1(k3_algstr_1)
    & v6_vectsp_1(k3_algstr_1)
    & v8_vectsp_1(k3_algstr_1)
    & v7_algstr_1(k3_algstr_1)
    & v8_algstr_1(k3_algstr_1) )).

fof(rc6_algstr_1,axiom,(
    ? [A] :
      ( l1_vectsp_1(A)
      & ~ v3_struct_0(A)
      & v2_group_1(A)
      & v4_group_1(A)
      & v7_group_1(A)
      & v1_vectsp_1(A)
      & v6_vectsp_1(A)
      & v8_vectsp_1(A)
      & v7_algstr_1(A)
      & v8_algstr_1(A) ) )).

fof(fc7_algstr_1,axiom,
    ( ~ v3_struct_0(k6_algstr_1)
    & v2_vectsp_1(k6_algstr_1) )).

fof(fc8_algstr_1,axiom,
    ( ~ v3_struct_0(k6_algstr_1)
    & v2_group_1(k6_algstr_1)
    & v2_vectsp_1(k6_algstr_1)
    & v6_vectsp_1(k6_algstr_1)
    & v8_vectsp_1(k6_algstr_1) )).

fof(cc5_algstr_1,axiom,(
    ! [A] :
      ( l2_vectsp_1(A)
     => ( ( ~ v3_struct_0(A)
          & v11_algstr_1(A) )
       => ( ~ v3_struct_0(A)
          & v9_algstr_1(A)
          & v10_algstr_1(A) ) ) ) )).

fof(rc7_algstr_1,axiom,(
    ? [A] :
      ( l2_vectsp_1(A)
      & ~ v3_struct_0(A)
      & v2_group_1(A)
      & v2_vectsp_1(A)
      & v6_vectsp_1(A)
      & v8_vectsp_1(A)
      & ~ v10_vectsp_1(A)
      & v9_algstr_1(A)
      & v10_algstr_1(A)
      & v11_algstr_1(A) ) )).

fof(fc9_algstr_1,axiom,
    ( ~ v3_struct_0(k6_algstr_1)
    & v2_group_1(k6_algstr_1)
    & v2_vectsp_1(k6_algstr_1)
    & v6_vectsp_1(k6_algstr_1)
    & v8_vectsp_1(k6_algstr_1)
    & v9_algstr_1(k6_algstr_1)
    & v10_algstr_1(k6_algstr_1)
    & v11_algstr_1(k6_algstr_1) )).

fof(rc8_algstr_1,axiom,(
    ? [A] :
      ( l2_vectsp_1(A)
      & ~ v3_struct_0(A)
      & v2_group_1(A)
      & v4_group_1(A)
      & v2_vectsp_1(A)
      & v6_vectsp_1(A)
      & v8_vectsp_1(A)
      & ~ v10_vectsp_1(A)
      & v9_algstr_1(A)
      & v10_algstr_1(A)
      & v11_algstr_1(A) ) )).

fof(fc10_algstr_1,axiom,
    ( ~ v3_struct_0(k6_algstr_1)
    & v2_group_1(k6_algstr_1)
    & v4_group_1(k6_algstr_1)
    & v2_vectsp_1(k6_algstr_1)
    & v6_vectsp_1(k6_algstr_1)
    & v8_vectsp_1(k6_algstr_1)
    & v9_algstr_1(k6_algstr_1)
    & v10_algstr_1(k6_algstr_1)
    & v11_algstr_1(k6_algstr_1) )).

fof(rc9_algstr_1,axiom,(
    ? [A] :
      ( l2_vectsp_1(A)
      & ~ v3_struct_0(A)
      & v2_group_1(A)
      & v4_group_1(A)
      & v7_group_1(A)
      & v2_vectsp_1(A)
      & v6_vectsp_1(A)
      & v8_vectsp_1(A)
      & ~ v10_vectsp_1(A)
      & v9_algstr_1(A)
      & v10_algstr_1(A)
      & v11_algstr_1(A) ) )).

fof(t1_algstr_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_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,D,k1_rlvect_1(A)) = D )
                  & ! [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                     => ? [E] :
                          ( m1_subset_1(E,u1_struct_0(A))
                          & k2_rlvect_1(A,D,E) = k1_rlvect_1(A) ) )
                  & ! [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                     => ! [E] :
                          ( m1_subset_1(E,u1_struct_0(A))
                         => ! [F] :
                              ( m1_subset_1(F,u1_struct_0(A))
                             => k2_rlvect_1(A,k2_rlvect_1(A,D,E),F) = k2_rlvect_1(A,D,k2_rlvect_1(A,E,F)) ) ) )
                  & k2_rlvect_1(A,B,C) = k1_rlvect_1(A) )
               => k2_rlvect_1(A,C,B) = k1_rlvect_1(A) ) ) ) ) )).

fof(t2_algstr_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( ( ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => k2_rlvect_1(A,C,k1_rlvect_1(A)) = C )
              & ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ? [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                      & k2_rlvect_1(A,C,D) = k1_rlvect_1(A) ) )
              & ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ! [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                     => ! [E] :
                          ( m1_subset_1(E,u1_struct_0(A))
                         => k2_rlvect_1(A,k2_rlvect_1(A,C,D),E) = k2_rlvect_1(A,C,k2_rlvect_1(A,D,E)) ) ) ) )
           => k2_rlvect_1(A,k1_rlvect_1(A),B) = k2_rlvect_1(A,B,k1_rlvect_1(A)) ) ) ) )).

fof(t3_algstr_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_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,C,B) = k1_rlvect_1(A) ) ) ) ) )).

fof(d1_algstr_1,axiom,(
    $true )).

fof(d2_algstr_1,axiom,(
    $true )).

fof(d3_algstr_1,axiom,(
    ! [A] : k1_algstr_1(A) = A )).

fof(d4_algstr_1,axiom,(
    k2_algstr_1 = g1_rlvect_1(k1_tarski(k1_xboole_0),k2_midsp_1,k1_algstr_1(k1_xboole_0)) )).

fof(t4_algstr_1,axiom,(
    $true )).

fof(t5_algstr_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k2_algstr_1))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k2_algstr_1))
         => A = B ) ) )).

fof(t6_algstr_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k2_algstr_1))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k2_algstr_1))
         => k2_rlvect_1(k2_algstr_1,A,B) = k1_rlvect_1(k2_algstr_1) ) ) )).

fof(d5_algstr_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A) )
     => ( v1_algstr_1(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => k2_rlvect_1(A,k1_rlvect_1(A),B) = B ) ) ) )).

fof(d6_algstr_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A) )
     => ( v2_algstr_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,B,C) = k2_rlvect_1(A,B,D)
                     => C = D ) ) ) ) ) ) )).

fof(d7_algstr_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A) )
     => ( v3_algstr_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,C,B) = k2_rlvect_1(A,D,B)
                     => C = D ) ) ) ) ) ) )).

fof(d8_algstr_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A) )
     => ( v4_algstr_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,D,B) = C ) ) ) ) ) )).

fof(d9_algstr_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A) )
     => ( v5_algstr_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,B,D) = C ) ) ) ) ) )).

fof(d10_algstr_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A) )
     => ( v6_algstr_1(A)
      <=> ( v2_algstr_1(A)
          & v3_algstr_1(A)
          & v4_algstr_1(A)
          & v5_algstr_1(A) ) ) ) )).

fof(t7_algstr_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A) )
     => ( v6_algstr_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,B,D) = 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))
                      & k2_rlvect_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))
                     => ( k2_rlvect_1(A,B,C) = k2_rlvect_1(A,B,D)
                       => 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))
                     => ( k2_rlvect_1(A,C,B) = k2_rlvect_1(A,D,B)
                       => C = D ) ) ) ) ) ) ) )).

fof(t8_algstr_1,axiom,(
    $true )).

fof(t9_algstr_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A) )
     => ( ( ~ v3_struct_0(A)
          & v4_rlvect_1(A)
          & v5_rlvect_1(A)
          & v1_algstr_1(A)
          & v6_algstr_1(A)
          & l1_rlvect_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)) ) ) ) ) ) ) )).

fof(t10_algstr_1,axiom,(
    $true )).

fof(t11_algstr_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A) )
     => ( ( ~ v3_struct_0(A)
          & v3_rlvect_1(A)
          & v4_rlvect_1(A)
          & v5_rlvect_1(A)
          & v1_algstr_1(A)
          & v6_algstr_1(A)
          & l1_rlvect_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) ) ) ) ) ) )).

fof(d11_algstr_1,axiom,(
    k3_algstr_1 = g1_vectsp_1(k1_tarski(k1_xboole_0),k2_midsp_1,k1_algstr_1(k1_xboole_0)) )).

fof(t12_algstr_1,axiom,(
    $true )).

fof(t13_algstr_1,axiom,(
    $true )).

fof(t14_algstr_1,axiom,(
    $true )).

fof(t15_algstr_1,axiom,(
    $true )).

fof(t16_algstr_1,axiom,(
    $true )).

fof(t17_algstr_1,axiom,(
    $true )).

fof(t18_algstr_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k3_algstr_1))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k3_algstr_1))
         => A = B ) ) )).

fof(t19_algstr_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k3_algstr_1))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k3_algstr_1))
         => k1_group_1(k3_algstr_1,A,B) = k2_group_1(k3_algstr_1) ) ) )).

fof(d12_algstr_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A) )
     => ( v7_algstr_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,D) = 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,D,B) = C ) ) ) ) ) ) )).

fof(d13_algstr_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A) )
     => ( v8_algstr_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,C) = k1_group_1(A,B,D)
                       => 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)
                       => C = D ) ) ) ) ) ) ) )).

fof(t20_algstr_1,axiom,(
    $true )).

fof(t21_algstr_1,axiom,(
    $true )).

fof(t22_algstr_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A) )
     => ( ( ~ v3_struct_0(A)
          & v2_group_1(A)
          & v4_group_1(A)
          & v7_algstr_1(A)
          & v8_algstr_1(A)
          & l1_vectsp_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))
             => ? [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))
             => ! [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)) ) ) ) ) ) ) )).

fof(t23_algstr_1,axiom,(
    $true )).

fof(t24_algstr_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A) )
     => ( ( ~ v3_struct_0(A)
          & v2_group_1(A)
          & v4_group_1(A)
          & v7_group_1(A)
          & v7_algstr_1(A)
          & v8_algstr_1(A)
          & l1_vectsp_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))
             => ? [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))
             => ! [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))
                 => k1_group_1(A,B,C) = k1_group_1(A,C,B) ) ) ) ) ) )).

fof(d14_algstr_1,axiom,(
    $true )).

fof(d15_algstr_1,axiom,(
    $true )).

fof(d16_algstr_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v7_algstr_1(A)
        & v8_algstr_1(A)
        & l1_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( C = k4_algstr_1(A,B)
              <=> k1_group_1(A,B,C) = k2_group_1(A) ) ) ) ) )).

fof(t25_algstr_1,axiom,(
    $true )).

fof(t26_algstr_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & v4_group_1(A)
        & v7_algstr_1(A)
        & v8_algstr_1(A)
        & l1_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( k1_group_1(A,B,k4_algstr_1(A,B)) = k2_group_1(A)
            & k1_group_1(A,k4_algstr_1(A,B),B) = k2_group_1(A) ) ) ) )).

fof(d17_algstr_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v7_algstr_1(A)
        & v8_algstr_1(A)
        & l1_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => k5_algstr_1(A,B,C) = k1_group_1(A,B,k4_algstr_1(A,C)) ) ) ) )).

fof(d18_algstr_1,axiom,(
    $true )).

fof(d19_algstr_1,axiom,(
    $true )).

fof(d20_algstr_1,axiom,(
    $true )).

fof(d21_algstr_1,axiom,(
    k6_algstr_1 = g2_vectsp_1(k1_numbers,k35_binop_2,np__1,np__0) )).

fof(t27_algstr_1,axiom,(
    $true )).

fof(t28_algstr_1,axiom,(
    $true )).

fof(t29_algstr_1,axiom,(
    $true )).

fof(t30_algstr_1,axiom,(
    $true )).

fof(t31_algstr_1,axiom,(
    $true )).

fof(t32_algstr_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ~ ( A != np__0
              & ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => B != k11_binop_2(A,C) ) ) ) ) )).

fof(t33_algstr_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ~ ( A != np__0
              & ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => B != k11_binop_2(C,A) ) ) ) ) )).

fof(d22_algstr_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_vectsp_1(A) )
     => ( v9_algstr_1(A)
      <=> ( ! [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 ) ) ) ) ) ) ) )).

fof(d23_algstr_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_vectsp_1(A) )
     => ( v10_algstr_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,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 ) ) ) ) ) ) ) ) )).

fof(d24_algstr_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_vectsp_1(A) )
     => ( v11_algstr_1(A)
      <=> ( v9_algstr_1(A)
          & v10_algstr_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) ) ) ) ) )).

fof(t34_algstr_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_vectsp_1(A) )
     => ( v11_algstr_1(A)
      <=> ( ! [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) ) ) ) ) )).

fof(t35_algstr_1,axiom,(
    $true )).

fof(t36_algstr_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_vectsp_1(A) )
     => ( ( ~ v3_struct_0(A)
          & v2_group_1(A)
          & v4_group_1(A)
          & ~ v10_vectsp_1(A)
          & v11_algstr_1(A)
          & l2_vectsp_1(A) )
      <=> ( 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))
             => ! [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))
             => 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) ) ) ) ) )).

fof(t37_algstr_1,axiom,(
    $true )).

fof(t38_algstr_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_vectsp_1(A) )
     => ( ( ~ v3_struct_0(A)
          & v2_group_1(A)
          & v4_group_1(A)
          & v7_group_1(A)
          & ~ v10_vectsp_1(A)
          & v11_algstr_1(A)
          & l2_vectsp_1(A) )
      <=> ( 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))
             => ! [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))
             => 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))
                 => k1_group_1(A,B,C) = k1_group_1(A,C,B) ) ) ) ) ) )).

fof(d25_algstr_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v9_algstr_1(A)
        & v10_algstr_1(A)
        & l2_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( B != k1_rlvect_1(A)
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ( C = k7_algstr_1(A,B)
                <=> k1_group_1(A,B,C) = k2_group_1(A) ) ) ) ) ) )).

fof(t39_algstr_1,axiom,(
    $true )).

fof(t40_algstr_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & v4_group_1(A)
        & v9_algstr_1(A)
        & v10_algstr_1(A)
        & l2_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( B != k1_rlvect_1(A)
           => ( k1_group_1(A,B,k7_algstr_1(A,B)) = k2_group_1(A)
              & k1_group_1(A,k7_algstr_1(A,B),B) = k2_group_1(A) ) ) ) ) )).

fof(d26_algstr_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v9_algstr_1(A)
        & v10_algstr_1(A)
        & l2_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => k8_algstr_1(A,B,C) = k1_group_1(A,B,k7_algstr_1(A,C)) ) ) ) )).

fof(dt_k1_algstr_1,axiom,(
    ! [A] : m1_subset_1(k1_algstr_1(A),k1_tarski(A)) )).

fof(dt_k2_algstr_1,axiom,
    ( v1_rlvect_1(k2_algstr_1)
    & l1_rlvect_1(k2_algstr_1) )).

fof(dt_k3_algstr_1,axiom,
    ( v1_vectsp_1(k3_algstr_1)
    & l1_vectsp_1(k3_algstr_1) )).

fof(dt_k4_algstr_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v7_algstr_1(A)
        & v8_algstr_1(A)
        & l1_vectsp_1(A)
        & m1_subset_1(B,u1_struct_0(A)) )
     => m1_subset_1(k4_algstr_1(A,B),u1_struct_0(A)) ) )).

fof(dt_k5_algstr_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v7_algstr_1(A)
        & v8_algstr_1(A)
        & l1_vectsp_1(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => m1_subset_1(k5_algstr_1(A,B,C),u1_struct_0(A)) ) )).

fof(dt_k6_algstr_1,axiom,
    ( v2_vectsp_1(k6_algstr_1)
    & l2_vectsp_1(k6_algstr_1) )).

fof(dt_k7_algstr_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v9_algstr_1(A)
        & v10_algstr_1(A)
        & l2_vectsp_1(A)
        & m1_subset_1(B,u1_struct_0(A)) )
     => m1_subset_1(k7_algstr_1(A,B),u1_struct_0(A)) ) )).

fof(dt_k8_algstr_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v9_algstr_1(A)
        & v10_algstr_1(A)
        & l2_vectsp_1(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => m1_subset_1(k8_algstr_1(A,B,C),u1_struct_0(A)) ) )).
%------------------------------------------------------------------------------