% Mizar problem: t12_algstr_2,algstr_2,144,40 
fof(t12_algstr_2,conjecture,(
    ! [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)) ) ) ) ) ) ) ),
    inference(mizar_bg_added,[status(thm)],[cc1_algstr_1,cc2_algstr_1,cc3_algstr_1,cc3_vectsp_1,cc4_algstr_1,cc4_vectsp_1,cc5_algstr_1,d11_vectsp_1,d13_vectsp_1,d19_vectsp_1,d21_vectsp_1,d2_group_1,d5_algstr_1,d5_rlvect_1,d6_rlvect_1,d7_rlvect_1,dt_k1_group_1,dt_k1_rlvect_1,dt_k2_group_1,dt_k2_rlvect_1,dt_l1_group_1,dt_l1_rlvect_1,dt_l1_struct_0,dt_l1_vectsp_1,dt_l2_struct_0,dt_l2_vectsp_1,dt_l3_vectsp_1,dt_m1_subset_1,dt_u1_struct_0,existence_l1_group_1,existence_l1_rlvect_1,existence_l1_struct_0,existence_l1_vectsp_1,existence_l2_struct_0,existence_l2_vectsp_1,existence_l3_vectsp_1,existence_m1_subset_1,t34_algstr_1,t3_algstr_1,t7_algstr_1]),
    [file(algstr_2,t12_algstr_2)]).

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) ) ) ) ),
    file(algstr_1,cc1_algstr_1),
    []).

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) ) ) ) ),
    file(algstr_1,cc2_algstr_1),
    []).

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) ) ) ) ),
    file(algstr_1,cc3_algstr_1),
    []).

fof(cc3_vectsp_1,axiom,(
    ! [A] : 
      ( l1_vectsp_1(A)
     => ( ( ~ v3_struct_0(A)
          & v2_group_1(A) )
       => ( ~ v3_struct_0(A)
          & v6_vectsp_1(A)
          & v8_vectsp_1(A) ) ) ) ),
    file(vectsp_1,cc3_vectsp_1),
    []).

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) ) ) ) ),
    file(algstr_1,cc4_algstr_1),
    []).

fof(cc4_vectsp_1,axiom,(
    ! [A] : 
      ( l1_vectsp_1(A)
     => ( ( ~ v3_struct_0(A)
          & v6_vectsp_1(A)
          & v8_vectsp_1(A) )
       => ( ~ v3_struct_0(A)
          & v2_group_1(A) ) ) ) ),
    file(vectsp_1,cc4_vectsp_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) ) ) ) ),
    file(algstr_1,cc5_algstr_1),
    []).

fof(d11_vectsp_1,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ( v4_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_rlvect_1(A,C,D)) = k2_rlvect_1(A,k1_group_1(A,B,C),k1_group_1(A,B,D)) ) ) ) ) ) ),
    file(vectsp_1,d11_vectsp_1),
    []).

fof(d13_vectsp_1,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A) )
     => ( v6_vectsp_1(A)
      <=> ! [B] : 
            ( m1_subset_1(B,u1_struct_0(A))
           => k1_group_1(A,B,k2_group_1(A)) = B ) ) ) ),
    file(vectsp_1,d13_vectsp_1),
    []).

fof(d19_vectsp_1,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A) )
     => ( v8_vectsp_1(A)
      <=> ! [B] : 
            ( m1_subset_1(B,u1_struct_0(A))
           => k1_group_1(A,k2_group_1(A),B) = B ) ) ) ),
    file(vectsp_1,d19_vectsp_1),
    []).

fof(d21_vectsp_1,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l2_vectsp_1(A) )
     => ( v10_vectsp_1(A)
      <=> k1_rlvect_1(A) = k2_group_1(A) ) ) ),
    file(vectsp_1,d21_vectsp_1),
    []).

fof(d2_group_1,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v2_group_1(A)
      <=> ? [B] : 
            ( m1_subset_1(B,u1_struct_0(A))
            & ! [C] : 
                ( m1_subset_1(C,u1_struct_0(A))
               => ( k1_group_1(A,C,B) = C
                  & k1_group_1(A,B,C) = C ) ) ) ) ) ),
    file(group_1,d2_group_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 ) ) ) ),
    file(algstr_1,d5_algstr_1),
    []).

fof(d5_rlvect_1,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A) )
     => ( v3_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,B,C) = k2_rlvect_1(A,C,B) ) ) ) ) ),
    file(rlvect_1,d5_rlvect_1),
    []).

fof(d6_rlvect_1,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A) )
     => ( v4_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)) ) ) ) ) ) ),
    file(rlvect_1,d6_rlvect_1),
    []).

fof(d7_rlvect_1,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A) )
     => ( v5_rlvect_1(A)
      <=> ! [B] : 
            ( m1_subset_1(B,u1_struct_0(A))
           => k2_rlvect_1(A,B,k1_rlvect_1(A)) = B ) ) ) ),
    file(rlvect_1,d7_rlvect_1),
    []).

fof(dt_k1_group_1,axiom,(
    ! [A,B,C] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => m1_subset_1(k1_group_1(A,B,C),u1_struct_0(A)) ) ),
    file(group_1,k1_group_1),
    []).

fof(dt_k1_rlvect_1,axiom,(
    ! [A] : 
      ( l2_struct_0(A)
     => m1_subset_1(k1_rlvect_1(A),u1_struct_0(A)) ) ),
    file(rlvect_1,k1_rlvect_1),
    []).

fof(dt_k2_group_1,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => m1_subset_1(k2_group_1(A),u1_struct_0(A)) ) ),
    file(group_1,k2_group_1),
    []).

fof(dt_k2_rlvect_1,axiom,(
    ! [A,B,C] : 
      ( ( ~ v3_struct_0(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_rlvect_1(A,B,C),u1_struct_0(A)) ) ),
    file(rlvect_1,k2_rlvect_1),
    []).

fof(dt_l1_group_1,axiom,(
    ! [A] : 
      ( l1_group_1(A)
     => l1_struct_0(A) ) ),
    file(group_1,l1_group_1),
    []).

fof(dt_l1_rlvect_1,axiom,(
    ! [A] : 
      ( l1_rlvect_1(A)
     => l2_struct_0(A) ) ),
    file(rlvect_1,l1_rlvect_1),
    []).

fof(dt_l1_struct_0,axiom,(
    $true ),
    file(struct_0,l1_struct_0),
    []).

fof(dt_l1_vectsp_1,axiom,(
    ! [A] : 
      ( l1_vectsp_1(A)
     => l1_group_1(A) ) ),
    file(vectsp_1,l1_vectsp_1),
    []).

fof(dt_l2_struct_0,axiom,(
    ! [A] : 
      ( l2_struct_0(A)
     => l1_struct_0(A) ) ),
    file(struct_0,l2_struct_0),
    []).

fof(dt_l2_vectsp_1,axiom,(
    ! [A] : 
      ( l2_vectsp_1(A)
     => ( l1_vectsp_1(A)
        & l2_struct_0(A) ) ) ),
    file(vectsp_1,l2_vectsp_1),
    []).

fof(dt_l3_vectsp_1,axiom,(
    ! [A] : 
      ( l3_vectsp_1(A)
     => ( l1_rlvect_1(A)
        & l2_vectsp_1(A) ) ) ),
    file(vectsp_1,l3_vectsp_1),
    []).

fof(dt_m1_subset_1,axiom,(
    $true ),
    file(subset_1,m1_subset_1),
    []).

fof(dt_u1_struct_0,axiom,(
    $true ),
    file(struct_0,u1_struct_0),
    []).

fof(existence_l1_group_1,axiom,(
    ? [A] : l1_group_1(A) ),
    file(group_1,l1_group_1),
    []).

fof(existence_l1_rlvect_1,axiom,(
    ? [A] : l1_rlvect_1(A) ),
    file(rlvect_1,l1_rlvect_1),
    []).

fof(existence_l1_struct_0,axiom,(
    ? [A] : l1_struct_0(A) ),
    file(struct_0,l1_struct_0),
    []).

fof(existence_l1_vectsp_1,axiom,(
    ? [A] : l1_vectsp_1(A) ),
    file(vectsp_1,l1_vectsp_1),
    []).

fof(existence_l2_struct_0,axiom,(
    ? [A] : l2_struct_0(A) ),
    file(struct_0,l2_struct_0),
    []).

fof(existence_l2_vectsp_1,axiom,(
    ? [A] : l2_vectsp_1(A) ),
    file(vectsp_1,l2_vectsp_1),
    []).

fof(existence_l3_vectsp_1,axiom,(
    ? [A] : l3_vectsp_1(A) ),
    file(vectsp_1,l3_vectsp_1),
    []).

fof(existence_m1_subset_1,axiom,(
    ! [A] : 
    ? [B] : m1_subset_1(B,A) ),
    file(subset_1,m1_subset_1),
    []).

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) ) ) ) ) ),
    file(algstr_1,t34_algstr_1),
    []).

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) ) ) ) ) ),
    file(algstr_1,t3_algstr_1),
    []).

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 ) ) ) ) ) ) ) ),
    file(algstr_1,t7_algstr_1),
    []).