% Mizar problem: t11_ringcat1,ringcat1,465,20 
fof(t11_ringcat1,conjecture,(
    ! [A] : 
      ( ( v2_ringcat1(A)
        & v3_ringcat1(A)
        & l1_ringcat1(A) )
     => ! [B] : 
          ( ( v2_ringcat1(B)
            & v3_ringcat1(B)
            & l1_ringcat1(B) )
         => ( k1_ringcat1(B) = k2_ringcat1(A)
           => ( k1_ringcat1(k6_ringcat1(B,A)) = k1_ringcat1(A)
              & k2_ringcat1(k6_ringcat1(B,A)) = k2_ringcat1(B) ) ) ) ) ),
    inference(mizar_bg_added,[status(thm)],[abstractness_v2_ringcat1,antisymmetry_r2_hidden,cc1_relset_1,d3_ringcat1,d4_ringcat1,dt_g1_ringcat1,dt_k1_ringcat1,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_ringcat1,dt_k2_zfmisc_1,dt_k5_relat_1,dt_k6_ringcat1,dt_k7_funct_2,dt_l1_group_1,dt_l1_ringcat1,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_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_u1_ringcat1,dt_u1_struct_0,dt_u2_ringcat1,dt_u3_ringcat1,existence_l1_group_1,existence_l1_ringcat1,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_relset_1,existence_m1_subset_1,existence_m2_relset_1,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc4_subset_1,free_g1_ringcat1,rc1_ringcat1,rc1_subset_1,rc1_xboole_0,rc2_ringcat1,rc2_subset_1,rc2_xboole_0,rc3_struct_0,rc4_struct_0,rc5_struct_0,redefinition_k7_funct_2,redefinition_m2_relset_1,reflexivity_r1_tarski,t10_ringcat1,t1_subset,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t7_boole,t8_boole]),
    [file(ringcat1,t11_ringcat1)]).

fof(abstractness_v2_ringcat1,axiom,(
    ! [A] : 
      ( l1_ringcat1(A)
     => ( v2_ringcat1(A)
       => A = g1_ringcat1(u1_ringcat1(A),u2_ringcat1(A),u3_ringcat1(A)) ) ) ),
    file(ringcat1,v2_ringcat1),
    []).

fof(antisymmetry_r2_hidden,axiom,(
    ! [A,B] : 
      ( r2_hidden(A,B)
     => ~ r2_hidden(B,A) ) ),
    file(hidden,r2_hidden),
    []).

fof(cc1_relset_1,axiom,(
    ! [A,B,C] : 
      ( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B)))
     => v1_relat_1(C) ) ),
    file(relset_1,cc1_relset_1),
    []).

fof(d3_ringcat1,axiom,(
    ! [A] : 
      ( l1_ringcat1(A)
     => k1_ringcat1(A) = u1_ringcat1(A) ) ),
    file(ringcat1,d3_ringcat1),
    []).

fof(d4_ringcat1,axiom,(
    ! [A] : 
      ( l1_ringcat1(A)
     => k2_ringcat1(A) = u2_ringcat1(A) ) ),
    file(ringcat1,d4_ringcat1),
    []).

fof(dt_g1_ringcat1,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)
        & v4_group_1(B)
        & v6_vectsp_1(B)
        & v7_vectsp_1(B)
        & v8_vectsp_1(B)
        & l3_vectsp_1(B)
        & v1_funct_1(C)
        & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
        & m1_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
     => ( v2_ringcat1(g1_ringcat1(A,B,C))
        & l1_ringcat1(g1_ringcat1(A,B,C)) ) ) ),
    file(ringcat1,g1_ringcat1),
    []).

fof(dt_k1_ringcat1,axiom,(
    ! [A] : 
      ( l1_ringcat1(A)
     => ( ~ v3_struct_0(k1_ringcat1(A))
        & v3_rlvect_1(k1_ringcat1(A))
        & v4_rlvect_1(k1_ringcat1(A))
        & v5_rlvect_1(k1_ringcat1(A))
        & v6_rlvect_1(k1_ringcat1(A))
        & v4_group_1(k1_ringcat1(A))
        & v6_vectsp_1(k1_ringcat1(A))
        & v7_vectsp_1(k1_ringcat1(A))
        & v8_vectsp_1(k1_ringcat1(A))
        & l3_vectsp_1(k1_ringcat1(A)) ) ) ),
    file(ringcat1,k1_ringcat1),
    []).

fof(dt_k1_xboole_0,axiom,(
    $true ),
    file(xboole_0,k1_xboole_0),
    []).

fof(dt_k1_zfmisc_1,axiom,(
    $true ),
    file(zfmisc_1,k1_zfmisc_1),
    []).

fof(dt_k2_ringcat1,axiom,(
    ! [A] : 
      ( l1_ringcat1(A)
     => ( ~ v3_struct_0(k2_ringcat1(A))
        & v3_rlvect_1(k2_ringcat1(A))
        & v4_rlvect_1(k2_ringcat1(A))
        & v5_rlvect_1(k2_ringcat1(A))
        & v6_rlvect_1(k2_ringcat1(A))
        & v4_group_1(k2_ringcat1(A))
        & v6_vectsp_1(k2_ringcat1(A))
        & v7_vectsp_1(k2_ringcat1(A))
        & v8_vectsp_1(k2_ringcat1(A))
        & l3_vectsp_1(k2_ringcat1(A)) ) ) ),
    file(ringcat1,k2_ringcat1),
    []).

fof(dt_k2_zfmisc_1,axiom,(
    $true ),
    file(zfmisc_1,k2_zfmisc_1),
    []).

fof(dt_k5_relat_1,axiom,(
    ! [A,B] : 
      ( ( v1_relat_1(A)
        & v1_relat_1(B) )
     => v1_relat_1(k5_relat_1(A,B)) ) ),
    file(relat_1,k5_relat_1),
    []).

fof(dt_k6_ringcat1,axiom,(
    ! [A,B] : 
      ( ( v3_ringcat1(A)
        & l1_ringcat1(A)
        & v3_ringcat1(B)
        & l1_ringcat1(B) )
     => ( v2_ringcat1(k6_ringcat1(A,B))
        & v3_ringcat1(k6_ringcat1(A,B))
        & l1_ringcat1(k6_ringcat1(A,B)) ) ) ),
    file(ringcat1,k6_ringcat1),
    []).

fof(dt_k7_funct_2,axiom,(
    ! [A,B,C,D,E] : 
      ( ( ~ v1_xboole_0(B)
        & v1_funct_1(D)
        & v1_funct_2(D,A,B)
        & m1_relset_1(D,A,B)
        & v1_funct_1(E)
        & v1_funct_2(E,B,C)
        & m1_relset_1(E,B,C) )
     => ( v1_funct_1(k7_funct_2(A,B,C,D,E))
        & v1_funct_2(k7_funct_2(A,B,C,D,E),A,C)
        & m2_relset_1(k7_funct_2(A,B,C,D,E),A,C) ) ) ),
    file(funct_2,k7_funct_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_ringcat1,axiom,(
    $true ),
    file(ringcat1,l1_ringcat1),
    []).

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_relset_1,axiom,(
    $true ),
    file(relset_1,m1_relset_1),
    []).

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

fof(dt_m2_relset_1,axiom,(
    ! [A,B,C] : 
      ( m2_relset_1(C,A,B)
     => m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B))) ) ),
    file(relset_1,m2_relset_1),
    []).

fof(dt_u1_ringcat1,axiom,(
    ! [A] : 
      ( l1_ringcat1(A)
     => ( ~ v3_struct_0(u1_ringcat1(A))
        & v3_rlvect_1(u1_ringcat1(A))
        & v4_rlvect_1(u1_ringcat1(A))
        & v5_rlvect_1(u1_ringcat1(A))
        & v6_rlvect_1(u1_ringcat1(A))
        & v4_group_1(u1_ringcat1(A))
        & v6_vectsp_1(u1_ringcat1(A))
        & v7_vectsp_1(u1_ringcat1(A))
        & v8_vectsp_1(u1_ringcat1(A))
        & l3_vectsp_1(u1_ringcat1(A)) ) ) ),
    file(ringcat1,u1_ringcat1),
    []).

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

fof(dt_u2_ringcat1,axiom,(
    ! [A] : 
      ( l1_ringcat1(A)
     => ( ~ v3_struct_0(u2_ringcat1(A))
        & v3_rlvect_1(u2_ringcat1(A))
        & v4_rlvect_1(u2_ringcat1(A))
        & v5_rlvect_1(u2_ringcat1(A))
        & v6_rlvect_1(u2_ringcat1(A))
        & v4_group_1(u2_ringcat1(A))
        & v6_vectsp_1(u2_ringcat1(A))
        & v7_vectsp_1(u2_ringcat1(A))
        & v8_vectsp_1(u2_ringcat1(A))
        & l3_vectsp_1(u2_ringcat1(A)) ) ) ),
    file(ringcat1,u2_ringcat1),
    []).

fof(dt_u3_ringcat1,axiom,(
    ! [A] : 
      ( l1_ringcat1(A)
     => ( v1_funct_1(u3_ringcat1(A))
        & v1_funct_2(u3_ringcat1(A),u1_struct_0(u1_ringcat1(A)),u1_struct_0(u2_ringcat1(A)))
        & m2_relset_1(u3_ringcat1(A),u1_struct_0(u1_ringcat1(A)),u1_struct_0(u2_ringcat1(A))) ) ) ),
    file(ringcat1,u3_ringcat1),
    []).

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

fof(existence_l1_ringcat1,axiom,(
    ? [A] : l1_ringcat1(A) ),
    file(ringcat1,l1_ringcat1),
    []).

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_relset_1,axiom,(
    ! [A,B] : 
    ? [C] : m1_relset_1(C,A,B) ),
    file(relset_1,m1_relset_1),
    []).

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

fof(existence_m2_relset_1,axiom,(
    ! [A,B] : 
    ? [C] : m2_relset_1(C,A,B) ),
    file(relset_1,m2_relset_1),
    []).

fof(fc1_struct_0,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ~ v1_xboole_0(u1_struct_0(A)) ) ),
    file(struct_0,fc1_struct_0),
    []).

fof(fc1_subset_1,axiom,(
    ! [A] : ~ v1_xboole_0(k1_zfmisc_1(A)) ),
    file(subset_1,fc1_subset_1),
    []).

fof(fc1_xboole_0,axiom,(
    v1_xboole_0(k1_xboole_0) ),
    file(xboole_0,fc1_xboole_0),
    []).

fof(fc4_subset_1,axiom,(
    ! [A,B] : 
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B) )
     => ~ v1_xboole_0(k2_zfmisc_1(A,B)) ) ),
    file(subset_1,fc4_subset_1),
    []).

fof(free_g1_ringcat1,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)
        & v4_group_1(B)
        & v6_vectsp_1(B)
        & v7_vectsp_1(B)
        & v8_vectsp_1(B)
        & l3_vectsp_1(B)
        & v1_funct_1(C)
        & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
        & m1_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
     => ! [D,E,F] : 
          ( g1_ringcat1(A,B,C) = g1_ringcat1(D,E,F)
         => ( A = D
            & B = E
            & C = F ) ) ) ),
    file(ringcat1,g1_ringcat1),
    []).

fof(rc1_ringcat1,axiom,(
    ? [A] : 
      ( l1_ringcat1(A)
      & v2_ringcat1(A) ) ),
    file(ringcat1,rc1_ringcat1),
    []).

fof(rc1_subset_1,axiom,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ? [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
          & ~ v1_xboole_0(B) ) ) ),
    file(subset_1,rc1_subset_1),
    []).

fof(rc1_xboole_0,axiom,(
    ? [A] : v1_xboole_0(A) ),
    file(xboole_0,rc1_xboole_0),
    []).

fof(rc2_ringcat1,axiom,(
    ? [A] : 
      ( l1_ringcat1(A)
      & v2_ringcat1(A)
      & v3_ringcat1(A) ) ),
    file(ringcat1,rc2_ringcat1),
    []).

fof(rc2_subset_1,axiom,(
    ! [A] : 
    ? [B] : 
      ( m1_subset_1(B,k1_zfmisc_1(A))
      & v1_xboole_0(B) ) ),
    file(subset_1,rc2_subset_1),
    []).

fof(rc2_xboole_0,axiom,(
    ? [A] : ~ v1_xboole_0(A) ),
    file(xboole_0,rc2_xboole_0),
    []).

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(rc5_struct_0,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ? [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
          & ~ v1_xboole_0(B) ) ) ),
    file(struct_0,rc5_struct_0),
    []).

fof(redefinition_k7_funct_2,axiom,(
    ! [A,B,C,D,E] : 
      ( ( ~ v1_xboole_0(B)
        & v1_funct_1(D)
        & v1_funct_2(D,A,B)
        & m1_relset_1(D,A,B)
        & v1_funct_1(E)
        & v1_funct_2(E,B,C)
        & m1_relset_1(E,B,C) )
     => k7_funct_2(A,B,C,D,E) = k5_relat_1(D,E) ) ),
    file(funct_2,k7_funct_2),
    []).

fof(redefinition_m2_relset_1,axiom,(
    ! [A,B,C] : 
      ( m2_relset_1(C,A,B)
    <=> m1_relset_1(C,A,B) ) ),
    file(relset_1,m2_relset_1),
    []).

fof(reflexivity_r1_tarski,axiom,(
    ! [A,B] : r1_tarski(A,A) ),
    file(tarski,r1_tarski),
    []).

fof(t10_ringcat1,axiom,(
    ! [A] : 
      ( ( v2_ringcat1(A)
        & v3_ringcat1(A)
        & l1_ringcat1(A) )
     => ! [B] : 
          ( ( v2_ringcat1(B)
            & v3_ringcat1(B)
            & l1_ringcat1(B) )
         => ~ ( k1_ringcat1(B) = k2_ringcat1(A)
              & ! [C] : 
                  ( ( ~ v3_struct_0(C)
                    & v3_rlvect_1(C)
                    & v4_rlvect_1(C)
                    & v5_rlvect_1(C)
                    & v6_rlvect_1(C)
                    & v4_group_1(C)
                    & v6_vectsp_1(C)
                    & v7_vectsp_1(C)
                    & v8_vectsp_1(C)
                    & l3_vectsp_1(C) )
                 => ! [D] : 
                      ( ( ~ v3_struct_0(D)
                        & v3_rlvect_1(D)
                        & v4_rlvect_1(D)
                        & v5_rlvect_1(D)
                        & v6_rlvect_1(D)
                        & v4_group_1(D)
                        & v6_vectsp_1(D)
                        & v7_vectsp_1(D)
                        & v8_vectsp_1(D)
                        & l3_vectsp_1(D) )
                     => ! [E] : 
                          ( ( ~ v3_struct_0(E)
                            & v3_rlvect_1(E)
                            & v4_rlvect_1(E)
                            & v5_rlvect_1(E)
                            & v6_rlvect_1(E)
                            & v4_group_1(E)
                            & v6_vectsp_1(E)
                            & v7_vectsp_1(E)
                            & v8_vectsp_1(E)
                            & l3_vectsp_1(E) )
                         => ! [F] : 
                              ( ( v1_funct_1(F)
                                & v1_funct_2(F,u1_struct_0(C),u1_struct_0(D))
                                & m2_relset_1(F,u1_struct_0(C),u1_struct_0(D)) )
                             => ! [G] : 
                                  ( ( v1_funct_1(G)
                                    & v1_funct_2(G,u1_struct_0(D),u1_struct_0(E))
                                    & m2_relset_1(G,u1_struct_0(D),u1_struct_0(E)) )
                                 => ~ ( A = g1_ringcat1(C,D,F)
                                      & B = g1_ringcat1(D,E,G)
                                      & k6_ringcat1(B,A) = g1_ringcat1(C,E,k7_funct_2(u1_struct_0(C),u1_struct_0(D),u1_struct_0(E),F,G)) ) ) ) ) ) ) ) ) ) ),
    file(ringcat1,t10_ringcat1),
    []).

fof(t1_subset,axiom,(
    ! [A,B] : 
      ( r2_hidden(A,B)
     => m1_subset_1(A,B) ) ),
    file(subset,t1_subset),
    []).

fof(t2_subset,axiom,(
    ! [A,B] : 
      ( m1_subset_1(A,B)
     => ( v1_xboole_0(B)
        | r2_hidden(A,B) ) ) ),
    file(subset,t2_subset),
    []).

fof(t3_subset,axiom,(
    ! [A,B] : 
      ( m1_subset_1(A,k1_zfmisc_1(B))
    <=> r1_tarski(A,B) ) ),
    file(subset,t3_subset),
    []).

fof(t4_subset,axiom,(
    ! [A,B,C] : 
      ( ( r2_hidden(A,B)
        & m1_subset_1(B,k1_zfmisc_1(C)) )
     => m1_subset_1(A,C) ) ),
    file(subset,t4_subset),
    []).

fof(t5_subset,axiom,(
    ! [A,B,C] : ~ ( r2_hidden(A,B)
      & m1_subset_1(B,k1_zfmisc_1(C))
      & v1_xboole_0(C) ) ),
    file(subset,t5_subset),
    []).

fof(t6_boole,axiom,(
    ! [A] : 
      ( v1_xboole_0(A)
     => A = k1_xboole_0 ) ),
    file(boole,t6_boole),
    []).

fof(t7_boole,axiom,(
    ! [A,B] : ~ ( r2_hidden(A,B)
      & v1_xboole_0(B) ) ),
    file(boole,t7_boole),
    []).

fof(t8_boole,axiom,(
    ! [A,B] : ~ ( v1_xboole_0(A)
      & A != B
      & v1_xboole_0(B) ) ),
    file(boole,t8_boole),
    []).