SET007 Axioms: SET007+323.ax


%------------------------------------------------------------------------------
% File     : SET007+323 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Category of Rings
% 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    : ringcat1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  111 (   9 unit)
%            Number of atoms       : 1336 (  93 equality)
%            Maximal formula depth :   38 (  11 average)
%            Number of connectives : 1368 ( 143 ~  ;   0  |; 973  &)
%                                         (  25 <=>; 227 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   38 (   1 propositional; 0-4 arity)
%            Number of functors    :   50 (   1 constant; 0-6 arity)
%            Number of variables   :  272 (   0 singleton; 243 !;  29 ?)
%            Maximal term depth    :    4 (   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(rc1_ringcat1,axiom,(
    ? [A] :
      ( l1_ringcat1(A)
      & v2_ringcat1(A) ) )).

fof(fc1_ringcat1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ( v1_relat_1(k7_grcat_1(A))
        & v1_funct_1(k7_grcat_1(A))
        & v1_funct_2(k7_grcat_1(A),u1_struct_0(A),u1_struct_0(A))
        & v1_ringcat1(k7_grcat_1(A),A,A) ) ) )).

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

fof(fc2_ringcat1,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) )
     => ( v2_ringcat1(k4_ringcat1(A))
        & v3_ringcat1(k4_ringcat1(A)) ) ) )).

fof(rc3_ringcat1,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)
        & ~ 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) )
     => ? [C] :
          ( m1_ringcat1(C,A,B)
          & v2_ringcat1(C)
          & v3_ringcat1(C) ) ) )).

fof(rc4_ringcat1,axiom,(
    ? [A] :
      ( ~ v1_xboole_0(A)
      & v4_ringcat1(A) ) )).

fof(rc5_ringcat1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v4_ringcat1(A) )
     => ? [B] :
          ( m1_subset_1(B,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)
          & v3_vectsp_1(B)
          & v6_vectsp_1(B)
          & v7_vectsp_1(B)
          & v8_vectsp_1(B) ) ) )).

fof(rc6_ringcat1,axiom,(
    ? [A] :
      ( ~ v1_xboole_0(A)
      & v5_ringcat1(A) ) )).

fof(rc7_ringcat1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v5_ringcat1(A) )
     => ? [B] :
          ( m1_subset_1(B,A)
          & v2_ringcat1(B)
          & v3_ringcat1(B) ) ) )).

fof(rc8_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)
        & m4_ringcat1(C,A,B) )
     => ? [D] :
          ( m1_subset_1(D,C)
          & v2_ringcat1(D)
          & v3_ringcat1(D) ) ) )).

fof(fc3_ringcat1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ~ v1_xboole_0(k9_ringcat1(A)) ) )).

fof(fc4_ringcat1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ( ~ v1_xboole_0(k9_ringcat1(A))
        & v4_ringcat1(k9_ringcat1(A)) ) ) )).

fof(fc5_ringcat1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => v1_cat_1(k18_ringcat1(A)) ) )).

fof(fc6_ringcat1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ( v1_cat_1(k18_ringcat1(A))
        & v2_cat_1(k18_ringcat1(A)) ) ) )).

fof(d1_ringcat1,axiom,(
    $true )).

fof(d2_ringcat1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l3_vectsp_1(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( v1_ringcat1(C,A,B)
              <=> ( ! [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                     => ! [E] :
                          ( m1_subset_1(E,u1_struct_0(A))
                         => k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,k2_rlvect_1(A,D,E)) = k2_rlvect_1(B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,D),k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,E)) ) )
                  & ! [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                     => ! [E] :
                          ( m1_subset_1(E,u1_struct_0(A))
                         => k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,k1_group_1(A,D,E)) = k1_group_1(B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,D),k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,E)) ) )
                  & k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,k2_group_1(A)) = k2_group_1(B) ) ) ) ) ) )).

fof(t1_ringcat1,axiom,(
    $true )).

fof(t2_ringcat1,axiom,(
    $true )).

fof(t3_ringcat1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l3_vectsp_1(B) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & l3_vectsp_1(C) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,u1_struct_0(A),u1_struct_0(B))
                    & m2_relset_1(D,u1_struct_0(A),u1_struct_0(B)) )
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,u1_struct_0(B),u1_struct_0(C))
                        & m2_relset_1(E,u1_struct_0(B),u1_struct_0(C)) )
                     => ( ( v1_ringcat1(D,A,B)
                          & v1_ringcat1(E,B,C) )
                       => v1_ringcat1(k7_funct_2(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),D,E),A,C) ) ) ) ) ) ) )).

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

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

fof(d5_ringcat1,axiom,(
    ! [A] :
      ( l1_ringcat1(A)
     => k3_ringcat1(A) = u3_ringcat1(A) ) )).

fof(d6_ringcat1,axiom,(
    ! [A] :
      ( l1_ringcat1(A)
     => ( v3_ringcat1(A)
      <=> v1_ringcat1(k3_ringcat1(A),u1_ringcat1(A),u2_ringcat1(A)) ) ) )).

fof(d7_ringcat1,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) )
     => k4_ringcat1(A) = g1_ringcat1(A,A,k7_grcat_1(A)) ) )).

fof(d8_ringcat1,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)
            & v4_group_1(B)
            & v6_vectsp_1(B)
            & v7_vectsp_1(B)
            & v8_vectsp_1(B)
            & l3_vectsp_1(B) )
         => ( r1_ringcat1(A,B)
          <=> ? [C] :
                ( v3_ringcat1(C)
                & l1_ringcat1(C)
                & k1_ringcat1(C) = A
                & k2_ringcat1(C) = B ) ) ) ) )).

fof(d9_ringcat1,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)
            & v4_group_1(B)
            & v6_vectsp_1(B)
            & v7_vectsp_1(B)
            & v8_vectsp_1(B)
            & l3_vectsp_1(B) )
         => ( r1_ringcat1(A,B)
           => ! [C] :
                ( ( v2_ringcat1(C)
                  & v3_ringcat1(C)
                  & l1_ringcat1(C) )
               => ( m1_ringcat1(C,A,B)
                <=> ( k1_ringcat1(C) = A
                    & k2_ringcat1(C) = B ) ) ) ) ) ) )).

fof(t4_ringcat1,axiom,(
    $true )).

fof(t5_ringcat1,axiom,(
    ! [A] :
      ( ( v3_ringcat1(A)
        & l1_ringcat1(A) )
     => ! [B] :
          ( ( v3_ringcat1(B)
            & l1_ringcat1(B) )
         => ~ ( k1_ringcat1(A) = k2_ringcat1(B)
              & ! [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) )
                         => ~ ( r1_ringcat1(C,D)
                              & r1_ringcat1(D,E)
                              & m1_ringcat1(g1_ringcat1(u1_ringcat1(A),u2_ringcat1(A),u3_ringcat1(A)),D,E)
                              & m1_ringcat1(g1_ringcat1(u1_ringcat1(B),u2_ringcat1(B),u3_ringcat1(B)),C,D) ) ) ) ) ) ) ) )).

fof(t6_ringcat1,axiom,(
    ! [A] :
      ( ( v2_ringcat1(A)
        & v3_ringcat1(A)
        & l1_ringcat1(A) )
     => ( m1_ringcat1(A,k1_ringcat1(A),k2_ringcat1(A))
        & r1_ringcat1(k1_ringcat1(A),k2_ringcat1(A)) ) ) )).

fof(t7_ringcat1,axiom,(
    ! [A] :
      ( ( v2_ringcat1(A)
        & v3_ringcat1(A)
        & l1_ringcat1(A) )
     => ? [B] :
          ( ~ 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)
          & ? [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] :
                  ( v1_funct_1(D)
                  & v1_funct_2(D,u1_struct_0(B),u1_struct_0(C))
                  & m2_relset_1(D,u1_struct_0(B),u1_struct_0(C))
                  & m1_ringcat1(A,B,C)
                  & A = g1_ringcat1(B,C,D)
                  & v1_ringcat1(D,B,C) ) ) ) ) )).

fof(d10_ringcat1,axiom,(
    ! [A] :
      ( ( v3_ringcat1(A)
        & l1_ringcat1(A) )
     => ! [B] :
          ( ( v3_ringcat1(B)
            & l1_ringcat1(B) )
         => ( k1_ringcat1(A) = k2_ringcat1(B)
           => ! [C] :
                ( ( v2_ringcat1(C)
                  & v3_ringcat1(C)
                  & l1_ringcat1(C) )
               => ( C = k6_ringcat1(A,B)
                <=> ! [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] :
                              ( ( ~ v3_struct_0(F)
                                & v3_rlvect_1(F)
                                & v4_rlvect_1(F)
                                & v5_rlvect_1(F)
                                & v6_rlvect_1(F)
                                & v4_group_1(F)
                                & v6_vectsp_1(F)
                                & v7_vectsp_1(F)
                                & v8_vectsp_1(F)
                                & l3_vectsp_1(F) )
                             => ! [G] :
                                  ( ( v1_funct_1(G)
                                    & v1_funct_2(G,u1_struct_0(E),u1_struct_0(F))
                                    & m2_relset_1(G,u1_struct_0(E),u1_struct_0(F)) )
                                 => ! [H] :
                                      ( ( v1_funct_1(H)
                                        & v1_funct_2(H,u1_struct_0(D),u1_struct_0(E))
                                        & m2_relset_1(H,u1_struct_0(D),u1_struct_0(E)) )
                                     => ( ( g1_ringcat1(u1_ringcat1(A),u2_ringcat1(A),u3_ringcat1(A)) = g1_ringcat1(E,F,G)
                                          & g1_ringcat1(u1_ringcat1(B),u2_ringcat1(B),u3_ringcat1(B)) = g1_ringcat1(D,E,H) )
                                       => C = g1_ringcat1(D,F,k7_funct_2(u1_struct_0(D),u1_struct_0(E),u1_struct_0(F),H,G)) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t8_ringcat1,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)
            & v4_group_1(B)
            & v6_vectsp_1(B)
            & v7_vectsp_1(B)
            & v8_vectsp_1(B)
            & l3_vectsp_1(B) )
         => ! [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) )
             => ( ( r1_ringcat1(A,B)
                  & r1_ringcat1(B,C) )
               => r1_ringcat1(A,C) ) ) ) ) )).

fof(t9_ringcat1,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)
            & v4_group_1(B)
            & v6_vectsp_1(B)
            & v7_vectsp_1(B)
            & v8_vectsp_1(B)
            & l3_vectsp_1(B) )
         => ! [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] :
                  ( m1_ringcat1(D,A,B)
                 => ! [E] :
                      ( m1_ringcat1(E,C,A)
                     => ( ( r1_ringcat1(C,A)
                          & r1_ringcat1(A,B) )
                       => m1_ringcat1(k6_ringcat1(D,E),C,B) ) ) ) ) ) ) )).

fof(d11_ringcat1,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)
            & v4_group_1(B)
            & v6_vectsp_1(B)
            & v7_vectsp_1(B)
            & v8_vectsp_1(B)
            & l3_vectsp_1(B) )
         => ! [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] :
                  ( m1_ringcat1(D,B,C)
                 => ! [E] :
                      ( m1_ringcat1(E,A,B)
                     => ( ( r1_ringcat1(A,B)
                          & r1_ringcat1(B,C) )
                       => k7_ringcat1(A,B,C,D,E) = k6_ringcat1(D,E) ) ) ) ) ) ) )).

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)) ) ) ) ) ) ) ) ) ) )).

fof(t11_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)
           => ( k1_ringcat1(k6_ringcat1(B,A)) = k1_ringcat1(A)
              & k2_ringcat1(k6_ringcat1(B,A)) = k2_ringcat1(B) ) ) ) ) )).

fof(t12_ringcat1,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)
            & v4_group_1(B)
            & v6_vectsp_1(B)
            & v7_vectsp_1(B)
            & v8_vectsp_1(B)
            & l3_vectsp_1(B) )
         => ! [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] :
                      ( m1_ringcat1(E,A,B)
                     => ! [F] :
                          ( m1_ringcat1(F,B,C)
                         => ! [G] :
                              ( m1_ringcat1(G,C,D)
                             => ( ( r1_ringcat1(A,B)
                                  & r1_ringcat1(B,C)
                                  & r1_ringcat1(C,D) )
                               => k6_ringcat1(G,k6_ringcat1(F,E)) = k6_ringcat1(k6_ringcat1(G,F),E) ) ) ) ) ) ) ) ) )).

fof(t13_ringcat1,axiom,(
    ! [A] :
      ( ( v2_ringcat1(A)
        & v3_ringcat1(A)
        & l1_ringcat1(A) )
     => ! [B] :
          ( ( v2_ringcat1(B)
            & v3_ringcat1(B)
            & l1_ringcat1(B) )
         => ! [C] :
              ( ( v2_ringcat1(C)
                & v3_ringcat1(C)
                & l1_ringcat1(C) )
             => ( ( k1_ringcat1(C) = k2_ringcat1(B)
                  & k1_ringcat1(B) = k2_ringcat1(A) )
               => k6_ringcat1(C,k6_ringcat1(B,A)) = k6_ringcat1(k6_ringcat1(C,B),A) ) ) ) ) )).

fof(t14_ringcat1,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) )
     => ( k1_ringcat1(k5_ringcat1(A)) = A
        & k2_ringcat1(k5_ringcat1(A)) = A
        & ! [B] :
            ( ( v2_ringcat1(B)
              & v3_ringcat1(B)
              & l1_ringcat1(B) )
           => ( k2_ringcat1(B) = A
             => k6_ringcat1(k5_ringcat1(A),B) = B ) )
        & ! [B] :
            ( ( v2_ringcat1(B)
              & v3_ringcat1(B)
              & l1_ringcat1(B) )
           => ( k1_ringcat1(B) = A
             => k6_ringcat1(B,k5_ringcat1(A)) = B ) ) ) ) )).

fof(d12_ringcat1,axiom,(
    ! [A] :
      ( v4_ringcat1(A)
    <=> ! [B] :
          ( m1_subset_1(B,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)
            & v3_vectsp_1(B)
            & v6_vectsp_1(B)
            & v7_vectsp_1(B)
            & v8_vectsp_1(B)
            & l3_vectsp_1(B) ) ) ) )).

fof(d13_ringcat1,axiom,(
    ! [A] :
      ( v5_ringcat1(A)
    <=> ! [B] :
          ( r2_hidden(B,A)
         => ( v2_ringcat1(B)
            & v3_ringcat1(B)
            & l1_ringcat1(B) ) ) ) )).

fof(t15_ringcat1,axiom,(
    $true )).

fof(t16_ringcat1,axiom,(
    $true )).

fof(t17_ringcat1,axiom,(
    ! [A] :
      ( ( v2_ringcat1(A)
        & v3_ringcat1(A)
        & l1_ringcat1(A) )
     => ( ~ v1_xboole_0(k1_tarski(A))
        & v5_ringcat1(k1_tarski(A)) ) ) )).

fof(d14_ringcat1,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)
            & v4_group_1(B)
            & v6_vectsp_1(B)
            & v7_vectsp_1(B)
            & v8_vectsp_1(B)
            & l3_vectsp_1(B) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & v5_ringcat1(C) )
             => ( m4_ringcat1(C,A,B)
              <=> ! [D] :
                    ( m3_ringcat1(D,C)
                   => m1_ringcat1(D,A,B) ) ) ) ) ) )).

fof(t18_ringcat1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ 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) )
         => ! [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) )
             => ( m4_ringcat1(A,B,C)
              <=> ! [D] :
                    ( m1_subset_1(D,A)
                   => m1_ringcat1(D,B,C) ) ) ) ) ) )).

fof(t19_ringcat1,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)
            & v4_group_1(B)
            & v6_vectsp_1(B)
            & v7_vectsp_1(B)
            & v8_vectsp_1(B)
            & l3_vectsp_1(B) )
         => ! [C] :
              ( m1_ringcat1(C,A,B)
             => m4_ringcat1(k1_tarski(C),A,B) ) ) ) )).

fof(d15_ringcat1,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)
            & v4_group_1(B)
            & v6_vectsp_1(B)
            & v7_vectsp_1(B)
            & v8_vectsp_1(B)
            & l3_vectsp_1(B) )
         => ( r1_ringcat1(A,B)
           => ! [C] :
                ( m4_ringcat1(C,A,B)
               => ( C = k8_ringcat1(A,B)
                <=> ! [D] :
                      ( r2_hidden(D,C)
                    <=> m1_ringcat1(D,A,B) ) ) ) ) ) ) )).

fof(d16_ringcat1,axiom,(
    ! [A,B] :
      ( r2_ringcat1(A,B)
    <=> ? [C,D,E,F,G,H] :
          ( A = k3_mcart_1(k4_mcart_1(C,D,E,F),G,H)
          & ? [I] :
              ( ~ v3_struct_0(I)
              & v3_rlvect_1(I)
              & v4_rlvect_1(I)
              & v5_rlvect_1(I)
              & v6_rlvect_1(I)
              & v4_group_1(I)
              & v3_vectsp_1(I)
              & v6_vectsp_1(I)
              & v7_vectsp_1(I)
              & v8_vectsp_1(I)
              & l3_vectsp_1(I)
              & B = I
              & C = u1_struct_0(I)
              & D = u1_rlvect_1(I)
              & E = k7_vectsp_1(I)
              & F = u2_struct_0(I)
              & G = u1_group_1(I)
              & H = u1_vectsp_1(I) ) ) ) )).

fof(t20_ringcat1,axiom,(
    ! [A,B,C] :
      ( ( r2_ringcat1(A,B)
        & r2_ringcat1(A,C) )
     => B = C ) )).

fof(t21_ringcat1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ? [B] :
          ( r2_hidden(B,A)
          & r2_ringcat1(B,k18_mod_2) ) ) )).

fof(d17_ringcat1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ! [B] :
          ( B = k9_ringcat1(A)
        <=> ! [C] :
              ( r2_hidden(C,B)
            <=> ? [D] :
                  ( r2_hidden(D,A)
                  & r2_ringcat1(D,C) ) ) ) ) )).

fof(t22_ringcat1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => r2_hidden(k18_mod_2,k9_ringcat1(A)) ) )).

fof(t23_ringcat1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ! [B] :
          ( m1_subset_1(B,k9_ringcat1(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)
            & v3_vectsp_1(B)
            & v6_vectsp_1(B)
            & v7_vectsp_1(B)
            & v8_vectsp_1(B)
            & l3_vectsp_1(B) ) ) ) )).

fof(d18_ringcat1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v4_ringcat1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v5_ringcat1(B) )
         => ( B = k10_ringcat1(A)
          <=> ! [C] :
                ( r2_hidden(C,B)
              <=> ? [D] :
                    ( m2_ringcat1(D,A)
                    & ? [E] :
                        ( m2_ringcat1(E,A)
                        & r1_ringcat1(D,E)
                        & m1_ringcat1(C,D,E) ) ) ) ) ) ) )).

fof(d19_ringcat1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v4_ringcat1(A) )
     => ! [B] :
          ( m2_ringcat1(B,A)
         => k13_ringcat1(A,B) = k5_ringcat1(B) ) ) )).

fof(d20_ringcat1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v4_ringcat1(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k10_ringcat1(A),A)
            & m2_relset_1(B,k10_ringcat1(A),A) )
         => ( B = k14_ringcat1(A)
          <=> ! [C] :
                ( m3_ringcat1(C,k10_ringcat1(A))
               => k8_funct_2(k10_ringcat1(A),A,B,C) = k11_ringcat1(A,C) ) ) ) ) )).

fof(d21_ringcat1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v4_ringcat1(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k10_ringcat1(A),A)
            & m2_relset_1(B,k10_ringcat1(A),A) )
         => ( B = k15_ringcat1(A)
          <=> ! [C] :
                ( m3_ringcat1(C,k10_ringcat1(A))
               => k8_funct_2(k10_ringcat1(A),A,B,C) = k12_ringcat1(A,C) ) ) ) ) )).

fof(d22_ringcat1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v4_ringcat1(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,A,k10_ringcat1(A))
            & m2_relset_1(B,A,k10_ringcat1(A)) )
         => ( B = k16_ringcat1(A)
          <=> ! [C] :
                ( m2_ringcat1(C,A)
               => k8_funct_2(A,k10_ringcat1(A),B,C) = k13_ringcat1(A,C) ) ) ) ) )).

fof(t24_ringcat1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v4_ringcat1(A) )
     => ! [B] :
          ( m3_ringcat1(B,k10_ringcat1(A))
         => ! [C] :
              ( m3_ringcat1(C,k10_ringcat1(A))
             => ~ ( k11_ringcat1(A,B) = k12_ringcat1(A,C)
                  & ! [D] :
                      ( m2_ringcat1(D,A)
                     => ! [E] :
                          ( m2_ringcat1(E,A)
                         => ! [F] :
                              ( m2_ringcat1(F,A)
                             => ~ ( r1_ringcat1(D,E)
                                  & r1_ringcat1(E,F)
                                  & m1_ringcat1(B,E,F)
                                  & m1_ringcat1(C,D,E) ) ) ) ) ) ) ) ) )).

fof(t25_ringcat1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v4_ringcat1(A) )
     => ! [B] :
          ( m3_ringcat1(B,k10_ringcat1(A))
         => ! [C] :
              ( m3_ringcat1(C,k10_ringcat1(A))
             => ( k11_ringcat1(A,B) = k12_ringcat1(A,C)
               => r2_hidden(k6_ringcat1(B,C),k10_ringcat1(A)) ) ) ) ) )).

fof(d23_ringcat1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v4_ringcat1(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k2_zfmisc_1(k10_ringcat1(A),k10_ringcat1(A)),k10_ringcat1(A)) )
         => ( B = k17_ringcat1(A)
          <=> ( ! [C] :
                  ( m3_ringcat1(C,k10_ringcat1(A))
                 => ! [D] :
                      ( m3_ringcat1(D,k10_ringcat1(A))
                     => ( r2_hidden(k4_tarski(C,D),k1_relat_1(B))
                      <=> k11_ringcat1(A,C) = k12_ringcat1(A,D) ) ) )
              & ! [C] :
                  ( m3_ringcat1(C,k10_ringcat1(A))
                 => ! [D] :
                      ( m3_ringcat1(D,k10_ringcat1(A))
                     => ( r2_hidden(k4_tarski(C,D),k1_relat_1(B))
                       => k1_funct_1(B,k4_tarski(C,D)) = k6_ringcat1(C,D) ) ) ) ) ) ) ) )).

fof(d24_ringcat1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => k18_ringcat1(A) = g1_cat_1(k9_ringcat1(A),k10_ringcat1(k9_ringcat1(A)),k14_ringcat1(k9_ringcat1(A)),k15_ringcat1(k9_ringcat1(A)),k17_ringcat1(k9_ringcat1(A)),k16_ringcat1(k9_ringcat1(A))) ) )).

fof(t26_ringcat1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ! [B] :
          ( m1_subset_1(B,u2_cat_1(k18_ringcat1(A)))
         => ! [C] :
              ( m1_subset_1(C,u2_cat_1(k18_ringcat1(A)))
             => ( r2_hidden(k4_tarski(C,B),k1_relat_1(u5_cat_1(k18_ringcat1(A))))
              <=> k2_cat_1(k18_ringcat1(A),C) = k3_cat_1(k18_ringcat1(A),B) ) ) ) ) )).

fof(t27_ringcat1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ! [B] :
          ( m1_subset_1(B,u2_cat_1(k18_ringcat1(A)))
         => ! [C] :
              ( m3_ringcat1(C,k10_ringcat1(k9_ringcat1(A)))
             => ! [D] :
                  ( m1_subset_1(D,u1_cat_1(k18_ringcat1(A)))
                 => ! [E] :
                      ( m2_ringcat1(E,k9_ringcat1(A))
                     => ( v2_ringcat1(B)
                        & m3_ringcat1(B,k10_ringcat1(k9_ringcat1(A)))
                        & m1_subset_1(C,u2_cat_1(k18_ringcat1(A)))
                        & v3_vectsp_1(D)
                        & m2_ringcat1(D,k9_ringcat1(A))
                        & m1_subset_1(E,u1_cat_1(k18_ringcat1(A))) ) ) ) ) ) ) )).

fof(t28_ringcat1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_cat_1(k18_ringcat1(A)))
         => ! [C] :
              ( m2_ringcat1(C,k9_ringcat1(A))
             => ( B = C
               => k5_cat_1(k18_ringcat1(A),B) = k13_ringcat1(k9_ringcat1(A),C) ) ) ) ) )).

fof(t29_ringcat1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ! [B] :
          ( m1_subset_1(B,u2_cat_1(k18_ringcat1(A)))
         => ! [C] :
              ( m3_ringcat1(C,k10_ringcat1(k9_ringcat1(A)))
             => ( B = C
               => ( k2_cat_1(k18_ringcat1(A),B) = k11_ringcat1(k9_ringcat1(A),C)
                  & k3_cat_1(k18_ringcat1(A),B) = k12_ringcat1(k9_ringcat1(A),C) ) ) ) ) ) )).

fof(t30_ringcat1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ! [B] :
          ( m1_subset_1(B,u2_cat_1(k18_ringcat1(A)))
         => ! [C] :
              ( m1_subset_1(C,u2_cat_1(k18_ringcat1(A)))
             => ! [D] :
                  ( m3_ringcat1(D,k10_ringcat1(k9_ringcat1(A)))
                 => ! [E] :
                      ( m3_ringcat1(E,k10_ringcat1(k9_ringcat1(A)))
                     => ( ( B = D
                          & C = E )
                       => ( ( k2_cat_1(k18_ringcat1(A),C) = k3_cat_1(k18_ringcat1(A),B)
                           => k11_ringcat1(k9_ringcat1(A),E) = k12_ringcat1(k9_ringcat1(A),D) )
                          & ( k11_ringcat1(k9_ringcat1(A),E) = k12_ringcat1(k9_ringcat1(A),D)
                           => k2_cat_1(k18_ringcat1(A),C) = k3_cat_1(k18_ringcat1(A),B) )
                          & ( k2_cat_1(k18_ringcat1(A),C) = k3_cat_1(k18_ringcat1(A),B)
                           => r2_hidden(k4_tarski(E,D),k1_relat_1(k17_ringcat1(k9_ringcat1(A)))) )
                          & ( r2_hidden(k4_tarski(E,D),k1_relat_1(k17_ringcat1(k9_ringcat1(A))))
                           => k2_cat_1(k18_ringcat1(A),C) = k3_cat_1(k18_ringcat1(A),B) )
                          & ( k2_cat_1(k18_ringcat1(A),C) = k3_cat_1(k18_ringcat1(A),B)
                           => k4_cat_1(k18_ringcat1(A),B,C) = k6_ringcat1(E,D) )
                          & ( k2_cat_1(k18_ringcat1(A),B) = k2_cat_1(k18_ringcat1(A),C)
                           => k11_ringcat1(k9_ringcat1(A),D) = k11_ringcat1(k9_ringcat1(A),E) )
                          & ( k11_ringcat1(k9_ringcat1(A),D) = k11_ringcat1(k9_ringcat1(A),E)
                           => k2_cat_1(k18_ringcat1(A),B) = k2_cat_1(k18_ringcat1(A),C) )
                          & ( k3_cat_1(k18_ringcat1(A),B) = k3_cat_1(k18_ringcat1(A),C)
                           => k12_ringcat1(k9_ringcat1(A),D) = k12_ringcat1(k9_ringcat1(A),E) )
                          & ( k12_ringcat1(k9_ringcat1(A),D) = k12_ringcat1(k9_ringcat1(A),E)
                           => k3_cat_1(k18_ringcat1(A),B) = k3_cat_1(k18_ringcat1(A),C) ) ) ) ) ) ) ) ) )).

fof(dt_m1_ringcat1,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)
        & ~ 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) )
     => ! [C] :
          ( m1_ringcat1(C,A,B)
         => ( v2_ringcat1(C)
            & v3_ringcat1(C)
            & l1_ringcat1(C) ) ) ) )).

fof(existence_m1_ringcat1,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)
        & ~ 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) )
     => ? [C] : m1_ringcat1(C,A,B) ) )).

fof(dt_m2_ringcat1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v4_ringcat1(A) )
     => ! [B] :
          ( m2_ringcat1(B,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) ) ) ) )).

fof(existence_m2_ringcat1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v4_ringcat1(A) )
     => ? [B] : m2_ringcat1(B,A) ) )).

fof(redefinition_m2_ringcat1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v4_ringcat1(A) )
     => ! [B] :
          ( m2_ringcat1(B,A)
        <=> m1_subset_1(B,A) ) ) )).

fof(dt_m3_ringcat1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v5_ringcat1(A) )
     => ! [B] :
          ( m3_ringcat1(B,A)
         => ( v3_ringcat1(B)
            & l1_ringcat1(B) ) ) ) )).

fof(existence_m3_ringcat1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v5_ringcat1(A) )
     => ? [B] : m3_ringcat1(B,A) ) )).

fof(redefinition_m3_ringcat1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v5_ringcat1(A) )
     => ! [B] :
          ( m3_ringcat1(B,A)
        <=> m1_subset_1(B,A) ) ) )).

fof(dt_m4_ringcat1,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)
        & ~ 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) )
     => ! [C] :
          ( m4_ringcat1(C,A,B)
         => ( ~ v1_xboole_0(C)
            & v5_ringcat1(C) ) ) ) )).

fof(existence_m4_ringcat1,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)
        & ~ 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) )
     => ? [C] : m4_ringcat1(C,A,B) ) )).

fof(dt_m5_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)
        & m4_ringcat1(C,A,B) )
     => ! [D] :
          ( m5_ringcat1(D,A,B,C)
         => m1_ringcat1(D,A,B) ) ) )).

fof(existence_m5_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)
        & m4_ringcat1(C,A,B) )
     => ? [D] : m5_ringcat1(D,A,B,C) ) )).

fof(redefinition_m5_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)
        & m4_ringcat1(C,A,B) )
     => ! [D] :
          ( m5_ringcat1(D,A,B,C)
        <=> m1_subset_1(D,C) ) ) )).

fof(dt_l1_ringcat1,axiom,(
    $true )).

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

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

fof(reflexivity_r1_ringcat1,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)
        & ~ 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) )
     => r1_ringcat1(A,A) ) )).

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)) ) ) )).

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)) ) ) )).

fof(dt_k3_ringcat1,axiom,(
    ! [A] :
      ( l1_ringcat1(A)
     => ( v1_funct_1(k3_ringcat1(A))
        & v1_funct_2(k3_ringcat1(A),u1_struct_0(u1_ringcat1(A)),u1_struct_0(u2_ringcat1(A)))
        & m2_relset_1(k3_ringcat1(A),u1_struct_0(u1_ringcat1(A)),u1_struct_0(u2_ringcat1(A))) ) ) )).

fof(dt_k4_ringcat1,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) )
     => ( v3_ringcat1(k4_ringcat1(A))
        & l1_ringcat1(k4_ringcat1(A)) ) ) )).

fof(dt_k5_ringcat1,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) )
     => ( v2_ringcat1(k5_ringcat1(A))
        & m1_ringcat1(k5_ringcat1(A),A,A) ) ) )).

fof(redefinition_k5_ringcat1,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) )
     => k5_ringcat1(A) = k4_ringcat1(A) ) )).

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)) ) ) )).

fof(dt_k7_ringcat1,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ 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)
        & ~ 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)
        & m1_ringcat1(D,B,C)
        & m1_ringcat1(E,A,B) )
     => ( v2_ringcat1(k7_ringcat1(A,B,C,D,E))
        & m1_ringcat1(k7_ringcat1(A,B,C,D,E),A,C) ) ) )).

fof(dt_k8_ringcat1,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)
        & ~ 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) )
     => m4_ringcat1(k8_ringcat1(A,B),A,B) ) )).

fof(dt_k9_ringcat1,axiom,(
    $true )).

fof(dt_k10_ringcat1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v4_ringcat1(A) )
     => ( ~ v1_xboole_0(k10_ringcat1(A))
        & v5_ringcat1(k10_ringcat1(A)) ) ) )).

fof(dt_k11_ringcat1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v4_ringcat1(A)
        & m1_subset_1(B,k10_ringcat1(A)) )
     => m2_ringcat1(k11_ringcat1(A,B),A) ) )).

fof(redefinition_k11_ringcat1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v4_ringcat1(A)
        & m1_subset_1(B,k10_ringcat1(A)) )
     => k11_ringcat1(A,B) = k1_ringcat1(B) ) )).

fof(dt_k12_ringcat1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v4_ringcat1(A)
        & m1_subset_1(B,k10_ringcat1(A)) )
     => m2_ringcat1(k12_ringcat1(A,B),A) ) )).

fof(redefinition_k12_ringcat1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v4_ringcat1(A)
        & m1_subset_1(B,k10_ringcat1(A)) )
     => k12_ringcat1(A,B) = k2_ringcat1(B) ) )).

fof(dt_k13_ringcat1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v4_ringcat1(A)
        & m1_subset_1(B,A) )
     => ( v2_ringcat1(k13_ringcat1(A,B))
        & m3_ringcat1(k13_ringcat1(A,B),k10_ringcat1(A)) ) ) )).

fof(dt_k14_ringcat1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v4_ringcat1(A) )
     => ( v1_funct_1(k14_ringcat1(A))
        & v1_funct_2(k14_ringcat1(A),k10_ringcat1(A),A)
        & m2_relset_1(k14_ringcat1(A),k10_ringcat1(A),A) ) ) )).

fof(dt_k15_ringcat1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v4_ringcat1(A) )
     => ( v1_funct_1(k15_ringcat1(A))
        & v1_funct_2(k15_ringcat1(A),k10_ringcat1(A),A)
        & m2_relset_1(k15_ringcat1(A),k10_ringcat1(A),A) ) ) )).

fof(dt_k16_ringcat1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v4_ringcat1(A) )
     => ( v1_funct_1(k16_ringcat1(A))
        & v1_funct_2(k16_ringcat1(A),A,k10_ringcat1(A))
        & m2_relset_1(k16_ringcat1(A),A,k10_ringcat1(A)) ) ) )).

fof(dt_k17_ringcat1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v4_ringcat1(A) )
     => ( v1_funct_1(k17_ringcat1(A))
        & m2_relset_1(k17_ringcat1(A),k2_zfmisc_1(k10_ringcat1(A),k10_ringcat1(A)),k10_ringcat1(A)) ) ) )).

fof(dt_k18_ringcat1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => l1_cat_1(k18_ringcat1(A)) ) )).

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)) ) ) )).

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)) ) ) )).

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))) ) ) )).

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)) ) ) )).

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 ) ) ) )).
%------------------------------------------------------------------------------