% Mizar problem: t18_mod_2,mod_2,386,35 
fof(t18_mod_2,conjecture,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] : 
          ( ( v3_mod_2(B,A)
            & l1_mod_2(B,A) )
         => ! [C] : 
              ( ( v3_mod_2(C,A)
                & l1_mod_2(C,A) )
             => ~ ( k2_mod_2(A,B) = k3_mod_2(A,C)
                  & ! [D] : 
                      ( ( ~ v3_struct_0(D)
                        & v3_rlvect_1(D)
                        & v4_rlvect_1(D)
                        & v5_rlvect_1(D)
                        & v6_rlvect_1(D)
                        & v12_vectsp_1(D,A)
                        & l4_vectsp_1(D,A) )
                     => ! [E] : 
                          ( ( ~ v3_struct_0(E)
                            & v3_rlvect_1(E)
                            & v4_rlvect_1(E)
                            & v5_rlvect_1(E)
                            & v6_rlvect_1(E)
                            & v12_vectsp_1(E,A)
                            & l4_vectsp_1(E,A) )
                         => ! [F] : 
                              ( ( ~ v3_struct_0(F)
                                & v3_rlvect_1(F)
                                & v4_rlvect_1(F)
                                & v5_rlvect_1(F)
                                & v6_rlvect_1(F)
                                & v12_vectsp_1(F,A)
                                & l4_vectsp_1(F,A) )
                             => ~ ( m1_mod_2(B,A,E,F)
                                  & m1_mod_2(C,A,D,E) ) ) ) ) ) ) ) ) ),
    inference(mizar_bg_added,[status(thm)],[cc1_algstr_1,cc1_vectsp_1,cc2_algstr_1,cc2_vectsp_1,cc3_algstr_1,cc3_vectsp_1,cc4_algstr_1,cc4_vectsp_1,d11_mod_2,d6_mod_2,d7_mod_2,dt_k2_mod_2,dt_k3_mod_2,dt_l1_group_1,dt_l1_mod_2,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_l4_vectsp_1,dt_m1_mod_2,dt_u1_mod_2,dt_u2_mod_2,existence_l1_group_1,existence_l1_mod_2,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_l4_vectsp_1,existence_m1_mod_2,rc3_struct_0,rc4_struct_0,t16_mod_2]),
    [file(mod_2,t18_mod_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(cc1_vectsp_1,axiom,(
    ! [A] : 
      ( l3_vectsp_1(A)
     => ( ( ~ v3_struct_0(A)
          & v7_vectsp_1(A) )
       => ( ~ v3_struct_0(A)
          & v4_vectsp_1(A)
          & v5_vectsp_1(A) ) ) ) ),
    file(vectsp_1,cc1_vectsp_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(cc2_vectsp_1,axiom,(
    ! [A] : 
      ( l3_vectsp_1(A)
     => ( ( ~ v3_struct_0(A)
          & v4_vectsp_1(A)
          & v5_vectsp_1(A) )
       => ( ~ v3_struct_0(A)
          & v7_vectsp_1(A) ) ) ) ),
    file(vectsp_1,cc2_vectsp_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(d11_mod_2,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & v3_rlvect_1(B)
            & v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & v12_vectsp_1(B,A)
            & l4_vectsp_1(B,A) )
         => ! [C] : 
              ( ( ~ v3_struct_0(C)
                & v3_rlvect_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v12_vectsp_1(C,A)
                & l4_vectsp_1(C,A) )
             => ! [D] : 
                  ( ( v3_mod_2(D,A)
                    & l1_mod_2(D,A) )
                 => ( m1_mod_2(D,A,B,C)
                  <=> ( k2_mod_2(A,D) = B
                      & k3_mod_2(A,D) = C ) ) ) ) ) ) ),
    file(mod_2,d11_mod_2),
    []).

fof(d6_mod_2,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] : 
          ( l1_mod_2(B,A)
         => k2_mod_2(A,B) = u1_mod_2(A,B) ) ) ),
    file(mod_2,d6_mod_2),
    []).

fof(d7_mod_2,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] : 
          ( l1_mod_2(B,A)
         => k3_mod_2(A,B) = u2_mod_2(A,B) ) ) ),
    file(mod_2,d7_mod_2),
    []).

fof(dt_k2_mod_2,axiom,(
    ! [A,B] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A)
        & l1_mod_2(B,A) )
     => ( ~ v3_struct_0(k2_mod_2(A,B))
        & v3_rlvect_1(k2_mod_2(A,B))
        & v4_rlvect_1(k2_mod_2(A,B))
        & v5_rlvect_1(k2_mod_2(A,B))
        & v6_rlvect_1(k2_mod_2(A,B))
        & v12_vectsp_1(k2_mod_2(A,B),A)
        & l4_vectsp_1(k2_mod_2(A,B),A) ) ) ),
    file(mod_2,k2_mod_2),
    []).

fof(dt_k3_mod_2,axiom,(
    ! [A,B] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A)
        & l1_mod_2(B,A) )
     => ( ~ v3_struct_0(k3_mod_2(A,B))
        & v3_rlvect_1(k3_mod_2(A,B))
        & v4_rlvect_1(k3_mod_2(A,B))
        & v5_rlvect_1(k3_mod_2(A,B))
        & v6_rlvect_1(k3_mod_2(A,B))
        & v12_vectsp_1(k3_mod_2(A,B),A)
        & l4_vectsp_1(k3_mod_2(A,B),A) ) ) ),
    file(mod_2,k3_mod_2),
    []).

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

fof(dt_l1_mod_2,axiom,(
    $true ),
    file(mod_2,l1_mod_2),
    []).

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_l4_vectsp_1,axiom,(
    ! [A] : 
      ( l1_struct_0(A)
     => ! [B] : 
          ( l4_vectsp_1(B,A)
         => l1_rlvect_1(B) ) ) ),
    file(vectsp_1,l4_vectsp_1),
    []).

fof(dt_m1_mod_2,axiom,(
    ! [A,B,C] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A)
        & ~ v3_struct_0(B)
        & v3_rlvect_1(B)
        & v4_rlvect_1(B)
        & v5_rlvect_1(B)
        & v6_rlvect_1(B)
        & v12_vectsp_1(B,A)
        & l4_vectsp_1(B,A)
        & ~ v3_struct_0(C)
        & v3_rlvect_1(C)
        & v4_rlvect_1(C)
        & v5_rlvect_1(C)
        & v6_rlvect_1(C)
        & v12_vectsp_1(C,A)
        & l4_vectsp_1(C,A) )
     => ! [D] : 
          ( m1_mod_2(D,A,B,C)
         => ( v3_mod_2(D,A)
            & l1_mod_2(D,A) ) ) ) ),
    file(mod_2,m1_mod_2),
    []).

fof(dt_u1_mod_2,axiom,(
    ! [A,B] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A)
        & l1_mod_2(B,A) )
     => ( ~ v3_struct_0(u1_mod_2(A,B))
        & v3_rlvect_1(u1_mod_2(A,B))
        & v4_rlvect_1(u1_mod_2(A,B))
        & v5_rlvect_1(u1_mod_2(A,B))
        & v6_rlvect_1(u1_mod_2(A,B))
        & v12_vectsp_1(u1_mod_2(A,B),A)
        & l4_vectsp_1(u1_mod_2(A,B),A) ) ) ),
    file(mod_2,u1_mod_2),
    []).

fof(dt_u2_mod_2,axiom,(
    ! [A,B] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A)
        & l1_mod_2(B,A) )
     => ( ~ v3_struct_0(u2_mod_2(A,B))
        & v3_rlvect_1(u2_mod_2(A,B))
        & v4_rlvect_1(u2_mod_2(A,B))
        & v5_rlvect_1(u2_mod_2(A,B))
        & v6_rlvect_1(u2_mod_2(A,B))
        & v12_vectsp_1(u2_mod_2(A,B),A)
        & l4_vectsp_1(u2_mod_2(A,B),A) ) ) ),
    file(mod_2,u2_mod_2),
    []).

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

fof(existence_l1_mod_2,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ? [B] : l1_mod_2(B,A) ) ),
    file(mod_2,l1_mod_2),
    []).

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_l4_vectsp_1,axiom,(
    ! [A] : 
      ( l1_struct_0(A)
     => ? [B] : l4_vectsp_1(B,A) ) ),
    file(vectsp_1,l4_vectsp_1),
    []).

fof(existence_m1_mod_2,axiom,(
    ! [A,B,C] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A)
        & ~ v3_struct_0(B)
        & v3_rlvect_1(B)
        & v4_rlvect_1(B)
        & v5_rlvect_1(B)
        & v6_rlvect_1(B)
        & v12_vectsp_1(B,A)
        & l4_vectsp_1(B,A)
        & ~ v3_struct_0(C)
        & v3_rlvect_1(C)
        & v4_rlvect_1(C)
        & v5_rlvect_1(C)
        & v6_rlvect_1(C)
        & v12_vectsp_1(C,A)
        & l4_vectsp_1(C,A) )
     => ? [D] : m1_mod_2(D,A,B,C) ) ),
    file(mod_2,m1_mod_2),
    []).

fof(rc3_struct_0,axiom,(
    ? [A] : 
      ( l1_struct_0(A)
      & ~ v3_struct_0(A) ) ),
    file(struct_0,rc3_struct_0),
    []).

fof(rc4_struct_0,axiom,(
    ? [A] : 
      ( l2_struct_0(A)
      & ~ v3_struct_0(A) ) ),
    file(struct_0,rc4_struct_0),
    []).

fof(t16_mod_2,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] : 
          ( ( v3_mod_2(B,A)
            & l1_mod_2(B,A) )
         => ? [C] : 
              ( ~ v3_struct_0(C)
              & v3_rlvect_1(C)
              & v4_rlvect_1(C)
              & v5_rlvect_1(C)
              & v6_rlvect_1(C)
              & v12_vectsp_1(C,A)
              & l4_vectsp_1(C,A)
              & ? [D] : 
                  ( ~ v3_struct_0(D)
                  & v3_rlvect_1(D)
                  & v4_rlvect_1(D)
                  & v5_rlvect_1(D)
                  & v6_rlvect_1(D)
                  & v12_vectsp_1(D,A)
                  & l4_vectsp_1(D,A)
                  & m1_mod_2(B,A,C,D) ) ) ) ) ),
    file(mod_2,t16_mod_2),
    []).