SET007 Axioms: SET007+777.ax


%------------------------------------------------------------------------------
% File     : SET007+777 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Characterization and Existence of Grobner Bases
% 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    : groeb_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   55 (   0 unit)
%            Number of atoms       : 1342 (  24 equality)
%            Maximal formula depth :   26 (  17 average)
%            Number of connectives : 1408 ( 121 ~  ;   2  |; 988  &)
%                                         (  10 <=>; 287 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   68 (   0 propositional; 1-6 arity)
%            Number of functors    :   31 (   3 constant; 0-4 arity)
%            Number of variables   :  267 (   0 singleton; 261 !;   6 ?)
%            Maximal term depth    :    5 (   1 average)
% SPC      : 

% Comments : The individual reference can be found in [Mat90] by looking for
%            the name provided by [Urb08].
%          : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
%          : These set theory axioms are used in encodings of problems in
%            various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(fc1_groeb_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => ( ~ v3_struct_0(g1_orders_2(k14_polynom1(A),k4_groeb_1(A)))
        & v1_orders_2(g1_orders_2(k14_polynom1(A),k4_groeb_1(A)))
        & v2_orders_2(g1_orders_2(k14_polynom1(A),k4_groeb_1(A)))
        & v3_orders_2(g1_orders_2(k14_polynom1(A),k4_groeb_1(A)))
        & v4_orders_2(g1_orders_2(k14_polynom1(A),k4_groeb_1(A)))
        & v4_dickson(g1_orders_2(k14_polynom1(A),k4_groeb_1(A))) ) ) )).

fof(t1_groeb_1,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
            & v1_relat_2(B)
            & v4_relat_2(B)
            & v6_relat_2(B)
            & v8_relat_2(B)
            & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
         => ! [C] :
              ( ( v2_group_1(C)
                & v4_group_1(C)
                & v7_group_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v7_vectsp_1(C)
                & v9_vectsp_1(C)
                & ~ v3_realset2(C)
                & l3_vectsp_1(C) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k14_polynom1(A),u1_struct_0(C))
                    & v2_polynom1(D,k14_polynom1(A),C)
                    & m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) )
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,k14_polynom1(A),u1_struct_0(C))
                        & v2_polynom1(E,k14_polynom1(A),C)
                        & m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) )
                     => ! [F] :
                          ( ( v1_funct_1(F)
                            & v1_funct_2(F,k14_polynom1(A),u1_struct_0(C))
                            & v2_polynom1(F,k14_polynom1(A),C)
                            & m2_relset_1(F,k14_polynom1(A),u1_struct_0(C)) )
                         => ~ ( r4_polyred(A,B,C,D,E,F)
                              & ! [G] :
                                  ( ( v1_funct_1(G)
                                    & v1_funct_2(G,k14_polynom1(A),u1_struct_0(C))
                                    & v3_polynom7(G,A,C)
                                    & m2_relset_1(G,k14_polynom1(A),u1_struct_0(C)) )
                                 => F != k25_polynom1(A,C,D,k28_polynom1(A,C,G,E)) ) ) ) ) ) ) ) ) )).

fof(t2_groeb_1,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
            & v1_relat_2(B)
            & v4_relat_2(B)
            & v6_relat_2(B)
            & v8_relat_2(B)
            & v2_bagorder(B,A)
            & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v2_group_1(C)
                & v4_group_1(C)
                & v7_group_1(C)
                & v3_rlvect_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v7_vectsp_1(C)
                & v9_vectsp_1(C)
                & ~ v10_vectsp_1(C)
                & l3_vectsp_1(C) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k14_polynom1(A),u1_struct_0(C))
                    & v2_polynom1(D,k14_polynom1(A),C)
                    & m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) )
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,k14_polynom1(A),u1_struct_0(C))
                        & v2_polynom1(E,k14_polynom1(A),C)
                        & m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) )
                     => ! [F] :
                          ( ( v1_funct_1(F)
                            & v1_funct_2(F,k14_polynom1(A),u1_struct_0(C))
                            & v2_polynom1(F,k14_polynom1(A),C)
                            & m2_relset_1(F,k14_polynom1(A),u1_struct_0(C)) )
                         => ~ ( r4_polyred(A,B,C,D,E,F)
                              & ! [G] :
                                  ( ( v1_funct_1(G)
                                    & v1_funct_2(G,k14_polynom1(A),u1_struct_0(C))
                                    & v3_polynom7(G,A,C)
                                    & m2_relset_1(G,k14_polynom1(A),u1_struct_0(C)) )
                                 => ~ ( F = k25_polynom1(A,C,D,k29_polynom1(A,C,G,E))
                                      & ~ r2_hidden(k3_termord(A,B,C,k29_polynom1(A,C,G,E)),k12_polynom1(k14_polynom1(A),C,F))
                                      & r1_termord(A,B,k3_termord(A,B,C,k29_polynom1(A,C,G,E)),k3_termord(A,B,C,D)) ) ) ) ) ) ) ) ) ) )).

fof(t3_groeb_1,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
            & v1_relat_2(B)
            & v4_relat_2(B)
            & v6_relat_2(B)
            & v8_relat_2(B)
            & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
         => ! [C] :
              ( ( v2_group_1(C)
                & v4_group_1(C)
                & v7_group_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v7_vectsp_1(C)
                & v9_vectsp_1(C)
                & ~ v3_realset2(C)
                & l3_vectsp_1(C) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k14_polynom1(A),u1_struct_0(C))
                    & v2_polynom1(D,k14_polynom1(A),C)
                    & m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) )
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,k14_polynom1(A),u1_struct_0(C))
                        & v2_polynom1(E,k14_polynom1(A),C)
                        & m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) )
                     => ! [F] :
                          ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
                         => ! [G] :
                              ( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
                             => ( ( r1_tarski(F,G)
                                  & r5_polyred(A,B,C,D,E,F) )
                               => r5_polyred(A,B,C,D,E,G) ) ) ) ) ) ) ) ) )).

fof(t4_groeb_1,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
            & v1_relat_2(B)
            & v4_relat_2(B)
            & v6_relat_2(B)
            & v8_relat_2(B)
            & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
         => ! [C] :
              ( ( v2_group_1(C)
                & v4_group_1(C)
                & v7_group_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v7_vectsp_1(C)
                & v9_vectsp_1(C)
                & ~ v3_realset2(C)
                & l3_vectsp_1(C) )
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
                 => ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
                     => ( r1_tarski(D,E)
                       => r1_tarski(k3_polyred(A,B,C,D),k3_polyred(A,B,C,E)) ) ) ) ) ) ) )).

fof(t5_groeb_1,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & l3_vectsp_1(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
                & v2_polynom1(C,k14_polynom1(A),B)
                & m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
             => k12_polynom1(k14_polynom1(A),B,k24_polynom1(A,B,C)) = k12_polynom1(k14_polynom1(A),B,C) ) ) ) )).

fof(t6_groeb_1,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
            & v1_relat_2(B)
            & v4_relat_2(B)
            & v6_relat_2(B)
            & v8_relat_2(B)
            & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v2_group_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v7_vectsp_1(C)
                & ~ v3_realset2(C)
                & l3_vectsp_1(C) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k14_polynom1(A),u1_struct_0(C))
                    & v2_polynom1(D,k14_polynom1(A),C)
                    & m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) )
                 => r6_pboole(A,k3_termord(A,B,C,k24_polynom1(A,C,D)),k3_termord(A,B,C,D)) ) ) ) ) )).

fof(t7_groeb_1,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
            & v1_relat_2(B)
            & v4_relat_2(B)
            & v6_relat_2(B)
            & v8_relat_2(B)
            & v2_bagorder(B,A)
            & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v2_group_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v7_vectsp_1(C)
                & ~ v3_realset2(C)
                & l3_vectsp_1(C) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k14_polynom1(A),u1_struct_0(C))
                    & v2_polynom1(D,k14_polynom1(A),C)
                    & m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) )
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,k14_polynom1(A),u1_struct_0(C))
                        & v2_polynom1(E,k14_polynom1(A),C)
                        & m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) )
                     => r1_termord(A,B,k3_termord(A,B,C,k25_polynom1(A,C,D,E)),k2_termord(A,B,k3_termord(A,B,C,D),k3_termord(A,B,C,E))) ) ) ) ) ) )).

fof(t8_groeb_1,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
            & v1_relat_2(B)
            & v4_relat_2(B)
            & v6_relat_2(B)
            & v8_relat_2(B)
            & v2_bagorder(B,A)
            & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
         => ! [C] :
              ( ( v2_group_1(C)
                & v4_group_1(C)
                & v7_group_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v7_vectsp_1(C)
                & v9_vectsp_1(C)
                & ~ v3_realset2(C)
                & l3_vectsp_1(C) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k14_polynom1(A),u1_struct_0(C))
                    & v2_polynom1(D,k14_polynom1(A),C)
                    & m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) )
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,k14_polynom1(A),u1_struct_0(C))
                        & v2_polynom1(E,k14_polynom1(A),C)
                        & m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) )
                     => ( r1_tarski(k12_polynom1(k14_polynom1(A),C,E),k12_polynom1(k14_polynom1(A),C,D))
                       => r1_polyred(A,B,C,E,D) ) ) ) ) ) ) )).

fof(t9_groeb_1,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
            & v1_relat_2(B)
            & v4_relat_2(B)
            & v6_relat_2(B)
            & v8_relat_2(B)
            & v2_bagorder(B,A)
            & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v2_group_1(C)
                & v4_group_1(C)
                & v7_group_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v7_vectsp_1(C)
                & v9_vectsp_1(C)
                & ~ v10_vectsp_1(C)
                & l3_vectsp_1(C) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k14_polynom1(A),u1_struct_0(C))
                    & v2_polynom1(D,k14_polynom1(A),C)
                    & v1_polynom7(D,A,C)
                    & m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) )
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,k14_polynom1(A),u1_struct_0(C))
                        & v2_polynom1(E,k14_polynom1(A),C)
                        & v1_polynom7(E,A,C)
                        & m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) )
                     => ( r6_polyred(A,B,C,D,E)
                       => r1_termord(A,B,k3_termord(A,B,C,E),k3_termord(A,B,C,D)) ) ) ) ) ) ) )).

fof(t10_groeb_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
            & v1_relat_2(B)
            & v4_relat_2(B)
            & v6_relat_2(B)
            & v8_relat_2(B)
            & v2_bagorder(B,A)
            & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
         => ! [C] :
              ( ( v2_group_1(C)
                & v4_group_1(C)
                & v7_group_1(C)
                & v3_rlvect_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v7_vectsp_1(C)
                & v9_vectsp_1(C)
                & ~ v3_realset2(C)
                & l3_vectsp_1(C) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k14_polynom1(A),u1_struct_0(C))
                    & v2_polynom1(D,k14_polynom1(A),C)
                    & m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) )
                 => v9_rewrite1(k3_polyred(A,B,C,k1_groeb_1(A,C,D))) ) ) ) ) )).

fof(t11_groeb_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
            & v1_relat_2(B)
            & v4_relat_2(B)
            & v6_relat_2(B)
            & v8_relat_2(B)
            & v2_bagorder(B,A)
            & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v2_group_1(C)
                & v4_group_1(C)
                & v7_group_1(C)
                & v3_rlvect_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v7_vectsp_1(C)
                & v9_vectsp_1(C)
                & ~ v10_vectsp_1(C)
                & l3_vectsp_1(C) )
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
                 => ( ? [E] :
                        ( v1_funct_1(E)
                        & v1_funct_2(E,k14_polynom1(A),u1_struct_0(C))
                        & v2_polynom1(E,k14_polynom1(A),C)
                        & m2_relset_1(E,k14_polynom1(A),u1_struct_0(C))
                        & r2_hidden(E,D)
                        & k7_ideal_1(k30_polynom1(A,C),D) = k7_ideal_1(k30_polynom1(A,C),k1_groeb_1(A,C,E)) )
                   => v9_rewrite1(k3_polyred(A,B,C,D)) ) ) ) ) ) )).

fof(t12_groeb_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
            & v1_relat_2(B)
            & v4_relat_2(B)
            & v6_relat_2(B)
            & v8_relat_2(B)
            & v2_bagorder(B,A)
            & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v2_group_1(C)
                & v4_group_1(C)
                & v7_group_1(C)
                & v3_rlvect_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v7_vectsp_1(C)
                & v9_vectsp_1(C)
                & ~ v10_vectsp_1(C)
                & l3_vectsp_1(C) )
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
                 => ( v9_rewrite1(k3_polyred(A,B,C,D))
                   => v7_rewrite1(k3_polyred(A,B,C,D)) ) ) ) ) ) )).

fof(t13_groeb_1,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
            & v1_relat_2(B)
            & v4_relat_2(B)
            & v6_relat_2(B)
            & v8_relat_2(B)
            & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
         => ! [C] :
              ( ( v2_group_1(C)
                & v4_group_1(C)
                & v7_group_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v7_vectsp_1(C)
                & v9_vectsp_1(C)
                & ~ v3_realset2(C)
                & l3_vectsp_1(C) )
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
                 => ( v7_rewrite1(k3_polyred(A,B,C,D))
                   => v4_rewrite1(k3_polyred(A,B,C,D)) ) ) ) ) ) )).

fof(t14_groeb_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
            & v1_relat_2(B)
            & v4_relat_2(B)
            & v6_relat_2(B)
            & v8_relat_2(B)
            & v2_bagorder(B,A)
            & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v2_group_1(C)
                & v4_group_1(C)
                & v7_group_1(C)
                & v3_rlvect_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v7_vectsp_1(C)
                & v9_vectsp_1(C)
                & ~ v10_vectsp_1(C)
                & l3_vectsp_1(C) )
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
                 => ( v4_rewrite1(k3_polyred(A,B,C,D))
                   => v8_rewrite1(k3_polyred(A,B,C,D)) ) ) ) ) ) )).

fof(t15_groeb_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
            & v1_relat_2(B)
            & v4_relat_2(B)
            & v6_relat_2(B)
            & v8_relat_2(B)
            & v2_bagorder(B,A)
            & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v2_group_1(C)
                & v4_group_1(C)
                & v7_group_1(C)
                & v3_rlvect_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v7_vectsp_1(C)
                & v9_vectsp_1(C)
                & ~ v10_vectsp_1(C)
                & l3_vectsp_1(C) )
             => ! [D] :
                  ( ( ~ v1_xboole_0(D)
                    & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
                 => ( v8_rewrite1(k3_polyred(A,B,C,D))
                   => ! [E] :
                        ( ( v1_funct_1(E)
                          & v1_funct_2(E,k14_polynom1(A),u1_struct_0(C))
                          & v2_polynom1(E,k14_polynom1(A),C)
                          & m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) )
                       => ( r2_hidden(E,k7_ideal_1(k30_polynom1(A,C),D))
                         => r1_rewrite1(k3_polyred(A,B,C,D),E,k26_polynom1(A,C)) ) ) ) ) ) ) ) )).

fof(t16_groeb_1,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
            & v1_relat_2(B)
            & v4_relat_2(B)
            & v6_relat_2(B)
            & v8_relat_2(B)
            & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
         => ! [C] :
              ( ( v2_group_1(C)
                & v4_group_1(C)
                & v7_group_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v7_vectsp_1(C)
                & v9_vectsp_1(C)
                & ~ v3_realset2(C)
                & l3_vectsp_1(C) )
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
                 => ( ! [E] :
                        ( ( v1_funct_1(E)
                          & v1_funct_2(E,k14_polynom1(A),u1_struct_0(C))
                          & v2_polynom1(E,k14_polynom1(A),C)
                          & m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) )
                       => ( r2_hidden(E,k7_ideal_1(k30_polynom1(A,C),D))
                         => r1_rewrite1(k3_polyred(A,B,C,D),E,k26_polynom1(A,C)) ) )
                   => ! [E] :
                        ( ( v1_funct_1(E)
                          & v1_funct_2(E,k14_polynom1(A),u1_struct_0(C))
                          & v2_polynom1(E,k14_polynom1(A),C)
                          & v1_polynom7(E,A,C)
                          & m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) )
                       => ( r2_hidden(E,k7_ideal_1(k30_polynom1(A,C),D))
                         => r7_polyred(A,B,C,E,D) ) ) ) ) ) ) ) )).

fof(t17_groeb_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
            & v1_relat_2(B)
            & v4_relat_2(B)
            & v6_relat_2(B)
            & v8_relat_2(B)
            & v2_bagorder(B,A)
            & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v2_group_1(C)
                & v4_group_1(C)
                & v7_group_1(C)
                & v3_rlvect_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v7_vectsp_1(C)
                & v9_vectsp_1(C)
                & ~ v10_vectsp_1(C)
                & l3_vectsp_1(C) )
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
                 => ( ! [E] :
                        ( ( v1_funct_1(E)
                          & v1_funct_2(E,k14_polynom1(A),u1_struct_0(C))
                          & v2_polynom1(E,k14_polynom1(A),C)
                          & v1_polynom7(E,A,C)
                          & m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) )
                       => ( r2_hidden(E,k7_ideal_1(k30_polynom1(A,C),D))
                         => r7_polyred(A,B,C,E,D) ) )
                   => ! [E] :
                        ( ( v1_funct_1(E)
                          & v1_funct_2(E,k14_polynom1(A),u1_struct_0(C))
                          & v2_polynom1(E,k14_polynom1(A),C)
                          & v1_polynom7(E,A,C)
                          & m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) )
                       => ( r2_hidden(E,k7_ideal_1(k30_polynom1(A,C),D))
                         => r10_polyred(A,B,C,E,D) ) ) ) ) ) ) ) )).

fof(t18_groeb_1,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
            & v1_relat_2(B)
            & v4_relat_2(B)
            & v6_relat_2(B)
            & v8_relat_2(B)
            & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
         => ! [C] :
              ( ( v2_group_1(C)
                & v4_group_1(C)
                & v7_group_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v7_vectsp_1(C)
                & v9_vectsp_1(C)
                & ~ v3_realset2(C)
                & l3_vectsp_1(C) )
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
                 => ( ! [E] :
                        ( ( v1_funct_1(E)
                          & v1_funct_2(E,k14_polynom1(A),u1_struct_0(C))
                          & v2_polynom1(E,k14_polynom1(A),C)
                          & v1_polynom7(E,A,C)
                          & m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) )
                       => ( r2_hidden(E,k7_ideal_1(k30_polynom1(A,C),D))
                         => r10_polyred(A,B,C,E,D) ) )
                   => ! [E] :
                        ( ( v7_seqm_3(E)
                          & v1_polynom1(E)
                          & m1_pboole(E,A) )
                       => ~ ( r2_hidden(E,k2_groeb_1(A,B,C,k7_ideal_1(k30_polynom1(A,C),D)))
                            & ! [F] :
                                ( ( v7_seqm_3(F)
                                  & v1_polynom1(F)
                                  & m1_pboole(F,A) )
                               => ~ ( r2_hidden(F,k2_groeb_1(A,B,C,D))
                                    & r3_polynom1(A,F,E) ) ) ) ) ) ) ) ) ) )).

fof(t19_groeb_1,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
            & v1_relat_2(B)
            & v4_relat_2(B)
            & v6_relat_2(B)
            & v8_relat_2(B)
            & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
         => ! [C] :
              ( ( v2_group_1(C)
                & v4_group_1(C)
                & v7_group_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v7_vectsp_1(C)
                & v9_vectsp_1(C)
                & ~ v3_realset2(C)
                & l3_vectsp_1(C) )
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
                 => ( ! [E] :
                        ( ( v7_seqm_3(E)
                          & v1_polynom1(E)
                          & m1_pboole(E,A) )
                       => ~ ( r2_hidden(E,k2_groeb_1(A,B,C,k7_ideal_1(k30_polynom1(A,C),D)))
                            & ! [F] :
                                ( ( v7_seqm_3(F)
                                  & v1_polynom1(F)
                                  & m1_pboole(F,A) )
                               => ~ ( r2_hidden(F,k2_groeb_1(A,B,C,D))
                                    & r3_polynom1(A,F,E) ) ) ) )
                   => r1_tarski(k2_groeb_1(A,B,C,k7_ideal_1(k30_polynom1(A,C),D)),k3_groeb_1(A,k2_groeb_1(A,B,C,D))) ) ) ) ) ) )).

fof(t20_groeb_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
            & v1_relat_2(B)
            & v4_relat_2(B)
            & v6_relat_2(B)
            & v8_relat_2(B)
            & v2_bagorder(B,A)
            & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v2_group_1(C)
                & v4_group_1(C)
                & v7_group_1(C)
                & v3_rlvect_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v7_vectsp_1(C)
                & v9_vectsp_1(C)
                & ~ v10_vectsp_1(C)
                & l3_vectsp_1(C) )
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
                 => ( r1_tarski(k2_groeb_1(A,B,C,k7_ideal_1(k30_polynom1(A,C),D)),k3_groeb_1(A,k2_groeb_1(A,B,C,D)))
                   => v9_rewrite1(k3_polyred(A,B,C,D)) ) ) ) ) ) )).

fof(d3_groeb_1,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
            & v1_relat_2(B)
            & v4_relat_2(B)
            & v6_relat_2(B)
            & v8_relat_2(B)
            & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
         => ! [C] :
              ( ( v2_group_1(C)
                & v4_group_1(C)
                & v7_group_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v7_vectsp_1(C)
                & v9_vectsp_1(C)
                & ~ v3_realset2(C)
                & l3_vectsp_1(C) )
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
                 => ( r1_groeb_1(A,B,C,D)
                  <=> v9_rewrite1(k3_polyred(A,B,C,D)) ) ) ) ) ) )).

fof(d4_groeb_1,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
            & v1_relat_2(B)
            & v4_relat_2(B)
            & v6_relat_2(B)
            & v8_relat_2(B)
            & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
         => ! [C] :
              ( ( v2_group_1(C)
                & v4_group_1(C)
                & v7_group_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v7_vectsp_1(C)
                & v9_vectsp_1(C)
                & ~ v3_realset2(C)
                & l3_vectsp_1(C) )
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
                 => ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
                     => ( r2_groeb_1(A,B,C,D,E)
                      <=> ( k7_ideal_1(k30_polynom1(A,C),D) = E
                          & v9_rewrite1(k3_polyred(A,B,C,D)) ) ) ) ) ) ) ) )).

fof(t21_groeb_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
            & v1_relat_2(B)
            & v4_relat_2(B)
            & v6_relat_2(B)
            & v8_relat_2(B)
            & v2_bagorder(B,A)
            & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v2_group_1(C)
                & v4_group_1(C)
                & v7_group_1(C)
                & v3_rlvect_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v7_vectsp_1(C)
                & v9_vectsp_1(C)
                & ~ v10_vectsp_1(C)
                & l3_vectsp_1(C) )
             => ! [D] :
                  ( ( ~ v1_xboole_0(D)
                    & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
                 => ! [E] :
                      ( ( ~ v1_xboole_0(E)
                        & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
                     => ( r2_groeb_1(A,B,C,D,E)
                       => m2_rewrite1(k3_polyred(A,B,C,D),k3_polyred(A,B,C,E)) ) ) ) ) ) ) )).

fof(t22_groeb_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
            & v1_relat_2(B)
            & v4_relat_2(B)
            & v6_relat_2(B)
            & v8_relat_2(B)
            & v2_bagorder(B,A)
            & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v2_group_1(C)
                & v4_group_1(C)
                & v7_group_1(C)
                & v3_rlvect_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v7_vectsp_1(C)
                & v9_vectsp_1(C)
                & ~ v10_vectsp_1(C)
                & l3_vectsp_1(C) )
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k30_polynom1(A,C)))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(k30_polynom1(A,C)))
                     => ! [F] :
                          ( ( ~ v1_xboole_0(F)
                            & m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
                         => ( r1_groeb_1(A,B,C,F)
                           => ( r11_polyred(k30_polynom1(A,C),k7_ideal_1(k30_polynom1(A,C),F),D,E)
                            <=> k2_rewrite1(k3_polyred(A,B,C,F),D) = k2_rewrite1(k3_polyred(A,B,C,F),E) ) ) ) ) ) ) ) ) )).

fof(t23_groeb_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
            & v1_relat_2(B)
            & v4_relat_2(B)
            & v6_relat_2(B)
            & v8_relat_2(B)
            & v2_bagorder(B,A)
            & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v2_group_1(C)
                & v4_group_1(C)
                & v7_group_1(C)
                & v3_rlvect_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v7_vectsp_1(C)
                & v9_vectsp_1(C)
                & ~ v10_vectsp_1(C)
                & l3_vectsp_1(C) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k14_polynom1(A),u1_struct_0(C))
                    & v2_polynom1(D,k14_polynom1(A),C)
                    & m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) )
                 => ! [E] :
                      ( ( ~ v1_xboole_0(E)
                        & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
                     => ( r1_groeb_1(A,B,C,E)
                       => ( r2_hidden(D,k7_ideal_1(k30_polynom1(A,C),E))
                        <=> r1_rewrite1(k3_polyred(A,B,C,E),D,k26_polynom1(A,C)) ) ) ) ) ) ) ) )).

fof(t24_groeb_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
            & v1_relat_2(B)
            & v4_relat_2(B)
            & v6_relat_2(B)
            & v8_relat_2(B)
            & v2_bagorder(B,A)
            & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v2_group_1(C)
                & v4_group_1(C)
                & v7_group_1(C)
                & v3_rlvect_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v7_vectsp_1(C)
                & v9_vectsp_1(C)
                & ~ v10_vectsp_1(C)
                & l3_vectsp_1(C) )
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
                 => ! [E] :
                      ( ( ~ v1_xboole_0(E)
                        & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
                     => ( r2_groeb_1(A,B,C,E,D)
                       => ! [F] :
                            ( ( v1_funct_1(F)
                              & v1_funct_2(F,k14_polynom1(A),u1_struct_0(C))
                              & v2_polynom1(F,k14_polynom1(A),C)
                              & m2_relset_1(F,k14_polynom1(A),u1_struct_0(C)) )
                           => ( r2_hidden(F,D)
                             => r1_rewrite1(k3_polyred(A,B,C,E),F,k26_polynom1(A,C)) ) ) ) ) ) ) ) ) )).

fof(t25_groeb_1,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
            & v1_relat_2(B)
            & v4_relat_2(B)
            & v6_relat_2(B)
            & v8_relat_2(B)
            & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
         => ! [C] :
              ( ( v2_group_1(C)
                & v4_group_1(C)
                & v7_group_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v7_vectsp_1(C)
                & v9_vectsp_1(C)
                & ~ v3_realset2(C)
                & l3_vectsp_1(C) )
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
                 => ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
                     => ( ! [F] :
                            ( ( v1_funct_1(F)
                              & v1_funct_2(F,k14_polynom1(A),u1_struct_0(C))
                              & v2_polynom1(F,k14_polynom1(A),C)
                              & m2_relset_1(F,k14_polynom1(A),u1_struct_0(C)) )
                           => ( r2_hidden(F,E)
                             => r1_rewrite1(k3_polyred(A,B,C,D),F,k26_polynom1(A,C)) ) )
                       => ! [F] :
                            ( ( v1_funct_1(F)
                              & v1_funct_2(F,k14_polynom1(A),u1_struct_0(C))
                              & v2_polynom1(F,k14_polynom1(A),C)
                              & v1_polynom7(F,A,C)
                              & m2_relset_1(F,k14_polynom1(A),u1_struct_0(C)) )
                           => ( r2_hidden(F,E)
                             => r7_polyred(A,B,C,F,D) ) ) ) ) ) ) ) ) )).

fof(t26_groeb_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
            & v1_relat_2(B)
            & v4_relat_2(B)
            & v6_relat_2(B)
            & v8_relat_2(B)
            & v2_bagorder(B,A)
            & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v2_group_1(C)
                & v4_group_1(C)
                & v7_group_1(C)
                & v3_rlvect_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v7_vectsp_1(C)
                & v9_vectsp_1(C)
                & ~ v10_vectsp_1(C)
                & l3_vectsp_1(C) )
             => ! [D] :
                  ( ( v1_ideal_1(D,k30_polynom1(A,C))
                    & v2_ideal_1(D,k30_polynom1(A,C))
                    & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
                 => ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
                     => ( ( r1_tarski(E,D)
                          & ! [F] :
                              ( ( v1_funct_1(F)
                                & v1_funct_2(F,k14_polynom1(A),u1_struct_0(C))
                                & v2_polynom1(F,k14_polynom1(A),C)
                                & v1_polynom7(F,A,C)
                                & m2_relset_1(F,k14_polynom1(A),u1_struct_0(C)) )
                             => ( r2_hidden(F,D)
                               => r7_polyred(A,B,C,F,E) ) ) )
                       => ! [F] :
                            ( ( v1_funct_1(F)
                              & v1_funct_2(F,k14_polynom1(A),u1_struct_0(C))
                              & v2_polynom1(F,k14_polynom1(A),C)
                              & v1_polynom7(F,A,C)
                              & m2_relset_1(F,k14_polynom1(A),u1_struct_0(C)) )
                           => ( r2_hidden(F,D)
                             => r10_polyred(A,B,C,F,E) ) ) ) ) ) ) ) ) )).

fof(t27_groeb_1,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
            & v1_relat_2(B)
            & v4_relat_2(B)
            & v6_relat_2(B)
            & v8_relat_2(B)
            & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
         => ! [C] :
              ( ( v2_group_1(C)
                & v4_group_1(C)
                & v7_group_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v7_vectsp_1(C)
                & v9_vectsp_1(C)
                & ~ v3_realset2(C)
                & l3_vectsp_1(C) )
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
                 => ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
                     => ( ! [F] :
                            ( ( v1_funct_1(F)
                              & v1_funct_2(F,k14_polynom1(A),u1_struct_0(C))
                              & v2_polynom1(F,k14_polynom1(A),C)
                              & v1_polynom7(F,A,C)
                              & m2_relset_1(F,k14_polynom1(A),u1_struct_0(C)) )
                           => ( r2_hidden(F,E)
                             => r10_polyred(A,B,C,F,D) ) )
                       => ! [F] :
                            ( ( v7_seqm_3(F)
                              & v1_polynom1(F)
                              & m1_pboole(F,A) )
                           => ~ ( r2_hidden(F,k2_groeb_1(A,B,C,E))
                                & ! [G] :
                                    ( ( v7_seqm_3(G)
                                      & v1_polynom1(G)
                                      & m1_pboole(G,A) )
                                   => ~ ( r2_hidden(G,k2_groeb_1(A,B,C,D))
                                        & r3_polynom1(A,G,F) ) ) ) ) ) ) ) ) ) ) )).

fof(t28_groeb_1,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
            & v1_relat_2(B)
            & v4_relat_2(B)
            & v6_relat_2(B)
            & v8_relat_2(B)
            & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
         => ! [C] :
              ( ( v2_group_1(C)
                & v4_group_1(C)
                & v7_group_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v7_vectsp_1(C)
                & v9_vectsp_1(C)
                & ~ v3_realset2(C)
                & l3_vectsp_1(C) )
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
                 => ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
                     => ( ! [F] :
                            ( ( v7_seqm_3(F)
                              & v1_polynom1(F)
                              & m1_pboole(F,A) )
                           => ~ ( r2_hidden(F,k2_groeb_1(A,B,C,E))
                                & ! [G] :
                                    ( ( v7_seqm_3(G)
                                      & v1_polynom1(G)
                                      & m1_pboole(G,A) )
                                   => ~ ( r2_hidden(G,k2_groeb_1(A,B,C,D))
                                        & r3_polynom1(A,G,F) ) ) ) )
                       => r1_tarski(k2_groeb_1(A,B,C,E),k3_groeb_1(A,k2_groeb_1(A,B,C,D))) ) ) ) ) ) ) )).

fof(t29_groeb_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
            & v1_relat_2(B)
            & v4_relat_2(B)
            & v6_relat_2(B)
            & v8_relat_2(B)
            & v2_bagorder(B,A)
            & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v2_group_1(C)
                & v4_group_1(C)
                & v7_group_1(C)
                & v3_rlvect_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v7_vectsp_1(C)
                & v9_vectsp_1(C)
                & ~ v10_vectsp_1(C)
                & l3_vectsp_1(C) )
             => ! [D] :
                  ( ( ~ v1_xboole_0(D)
                    & v1_ideal_1(D,k30_polynom1(A,C))
                    & v2_ideal_1(D,k30_polynom1(A,C))
                    & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
                 => ! [E] :
                      ( ( ~ v1_xboole_0(E)
                        & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
                     => ( ( r1_tarski(E,D)
                          & r1_tarski(k2_groeb_1(A,B,C,D),k3_groeb_1(A,k2_groeb_1(A,B,C,E))) )
                       => r2_groeb_1(A,B,C,E,D) ) ) ) ) ) ) )).

fof(t30_groeb_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
            & v1_relat_2(B)
            & v4_relat_2(B)
            & v6_relat_2(B)
            & v8_relat_2(B)
            & v2_bagorder(B,A)
            & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
         => ! [C] :
              ( ( v2_group_1(C)
                & v4_group_1(C)
                & v7_group_1(C)
                & v3_rlvect_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v7_vectsp_1(C)
                & v9_vectsp_1(C)
                & ~ v3_realset2(C)
                & l3_vectsp_1(C) )
             => r2_groeb_1(A,B,C,k1_groeb_1(A,C,k26_polynom1(A,C)),k1_groeb_1(A,C,k26_polynom1(A,C))) ) ) ) )).

fof(t31_groeb_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
            & v1_relat_2(B)
            & v4_relat_2(B)
            & v6_relat_2(B)
            & v8_relat_2(B)
            & v2_bagorder(B,A)
            & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
         => ! [C] :
              ( ( v2_group_1(C)
                & v4_group_1(C)
                & v7_group_1(C)
                & v3_rlvect_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v7_vectsp_1(C)
                & v9_vectsp_1(C)
                & ~ v3_realset2(C)
                & l3_vectsp_1(C) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k14_polynom1(A),u1_struct_0(C))
                    & v2_polynom1(D,k14_polynom1(A),C)
                    & m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) )
                 => r2_groeb_1(A,B,C,k1_groeb_1(A,C,D),k7_ideal_1(k30_polynom1(A,C),k1_groeb_1(A,C,D))) ) ) ) ) )).

fof(t32_groeb_1,axiom,(
    ! [A] :
      ( ( v1_partfun1(A,k14_polynom1(k1_xboole_0),k14_polynom1(k1_xboole_0))
        & v1_relat_2(A)
        & v4_relat_2(A)
        & v6_relat_2(A)
        & v8_relat_2(A)
        & v2_bagorder(A,k1_xboole_0)
        & m2_relset_1(A,k14_polynom1(k1_xboole_0),k14_polynom1(k1_xboole_0)) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_group_1(B)
            & v4_group_1(B)
            & v7_group_1(B)
            & v3_rlvect_1(B)
            & v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & v7_vectsp_1(B)
            & v9_vectsp_1(B)
            & ~ v10_vectsp_1(B)
            & l3_vectsp_1(B) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & v1_ideal_1(C,k30_polynom1(k1_xboole_0,B))
                & v2_ideal_1(C,k30_polynom1(k1_xboole_0,B))
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k30_polynom1(k1_xboole_0,B)))) )
             => ! [D] :
                  ( ( ~ v1_xboole_0(D)
                    & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(k1_xboole_0,B)))) )
                 => ( r1_tarski(D,C)
                   => ( D = k1_groeb_1(k1_xboole_0,B,k26_polynom1(k1_xboole_0,B))
                      | r2_groeb_1(k1_xboole_0,A,B,D,C) ) ) ) ) ) ) )).

fof(t33_groeb_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v3_ordinal1(A) )
     => ! [B] :
          ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
            & v1_relat_2(B)
            & v4_relat_2(B)
            & v6_relat_2(B)
            & v8_relat_2(B)
            & v2_bagorder(B,A)
            & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v2_group_1(C)
                & v4_group_1(C)
                & v7_group_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v7_vectsp_1(C)
                & v9_vectsp_1(C)
                & ~ v10_vectsp_1(C)
                & l3_vectsp_1(C) )
             => ~ ! [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
                   => r1_groeb_1(A,B,C,D) ) ) ) ) )).

fof(d5_groeb_1,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
            & v1_relat_2(B)
            & v4_relat_2(B)
            & v8_relat_2(B)
            & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
         => ( B = k4_groeb_1(A)
          <=> ! [C] :
                ( ( v7_seqm_3(C)
                  & v1_polynom1(C)
                  & m1_pboole(C,A) )
               => ! [D] :
                    ( ( v7_seqm_3(D)
                      & v1_polynom1(D)
                      & m1_pboole(D,A) )
                   => ( r2_hidden(k4_tarski(C,D),B)
                    <=> r3_polynom1(A,C,D) ) ) ) ) ) ) )).

fof(t34_groeb_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(g1_orders_2(k14_polynom1(A),k4_groeb_1(A)))))
         => ? [C] :
              ( v1_finset_1(C)
              & m1_subset_1(C,k1_zfmisc_1(k14_polynom1(A)))
              & r1_dickson(g1_orders_2(k14_polynom1(A),k4_groeb_1(A)),B,C) ) ) ) )).

fof(t35_groeb_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
            & v1_relat_2(B)
            & v4_relat_2(B)
            & v6_relat_2(B)
            & v8_relat_2(B)
            & v2_bagorder(B,A)
            & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v2_group_1(C)
                & v4_group_1(C)
                & v7_group_1(C)
                & v3_rlvect_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v7_vectsp_1(C)
                & v9_vectsp_1(C)
                & ~ v10_vectsp_1(C)
                & l3_vectsp_1(C) )
             => ! [D] :
                  ( ( ~ v1_xboole_0(D)
                    & v1_ideal_1(D,k30_polynom1(A,C))
                    & v2_ideal_1(D,k30_polynom1(A,C))
                    & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
                 => ? [E] :
                      ( v1_finset_1(E)
                      & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
                      & r2_groeb_1(A,B,C,E,D) ) ) ) ) ) )).

fof(t36_groeb_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
            & v1_relat_2(B)
            & v4_relat_2(B)
            & v6_relat_2(B)
            & v8_relat_2(B)
            & v2_bagorder(B,A)
            & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v2_group_1(C)
                & v4_group_1(C)
                & v7_group_1(C)
                & v3_rlvect_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v7_vectsp_1(C)
                & v9_vectsp_1(C)
                & ~ v10_vectsp_1(C)
                & l3_vectsp_1(C) )
             => ! [D] :
                  ( ( ~ v1_xboole_0(D)
                    & v1_ideal_1(D,k30_polynom1(A,C))
                    & v2_ideal_1(D,k30_polynom1(A,C))
                    & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
                 => ~ ( D != k1_groeb_1(A,C,k26_polynom1(A,C))
                      & ! [E] :
                          ( ( v1_finset_1(E)
                            & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
                         => ~ ( r2_groeb_1(A,B,C,E,D)
                              & ~ r2_hidden(k26_polynom1(A,C),E) ) ) ) ) ) ) ) )).

fof(d6_groeb_1,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
            & v1_relat_2(B)
            & v4_relat_2(B)
            & v6_relat_2(B)
            & v8_relat_2(B)
            & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & l2_vectsp_1(C) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k14_polynom1(A),u1_struct_0(C))
                    & v2_polynom1(D,k14_polynom1(A),C)
                    & m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) )
                 => ( r3_groeb_1(A,B,C,D)
                  <=> k4_termord(A,B,C,D) = k2_group_1(C) ) ) ) ) ) )).

fof(d7_groeb_1,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
            & v1_relat_2(B)
            & v4_relat_2(B)
            & v6_relat_2(B)
            & v8_relat_2(B)
            & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v2_group_1(C)
                & v4_group_1(C)
                & v7_group_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v7_vectsp_1(C)
                & v9_vectsp_1(C)
                & ~ v3_realset2(C)
                & l3_vectsp_1(C) )
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
                 => ( r4_groeb_1(A,B,C,D)
                  <=> ! [E] :
                        ( ( v1_funct_1(E)
                          & v1_funct_2(E,k14_polynom1(A),u1_struct_0(C))
                          & v2_polynom1(E,k14_polynom1(A),C)
                          & m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) )
                       => ( r2_hidden(E,D)
                         => ( r3_groeb_1(A,B,C,E)
                            & ~ r7_polyred(A,B,C,E,k6_subset_1(u1_struct_0(k30_polynom1(A,C)),D,k1_groeb_1(A,C,E))) ) ) ) ) ) ) ) ) )).

fof(t37_groeb_1,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
            & v1_relat_2(B)
            & v4_relat_2(B)
            & v6_relat_2(B)
            & v8_relat_2(B)
            & v2_bagorder(B,A)
            & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v2_group_1(C)
                & v4_group_1(C)
                & v7_group_1(C)
                & v3_rlvect_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v7_vectsp_1(C)
                & v9_vectsp_1(C)
                & ~ v10_vectsp_1(C)
                & l3_vectsp_1(C) )
             => ! [D] :
                  ( ( v1_ideal_1(D,k30_polynom1(A,C))
                    & v2_ideal_1(D,k30_polynom1(A,C))
                    & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,k14_polynom1(A),u1_struct_0(C))
                        & v3_polynom7(E,A,C)
                        & m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) )
                     => ! [F] :
                          ( ( v1_funct_1(F)
                            & v1_funct_2(F,k14_polynom1(A),u1_struct_0(C))
                            & v2_polynom1(F,k14_polynom1(A),C)
                            & m2_relset_1(F,k14_polynom1(A),u1_struct_0(C)) )
                         => ! [G] :
                              ( ( v1_funct_1(G)
                                & v1_funct_2(G,k14_polynom1(A),u1_struct_0(C))
                                & v2_polynom1(G,k14_polynom1(A),C)
                                & m2_relset_1(G,k14_polynom1(A),u1_struct_0(C)) )
                             => ( ( r2_hidden(F,D)
                                  & r2_hidden(G,D)
                                  & k5_termord(A,B,C,F) = E
                                  & k5_termord(A,B,C,G) = E
                                  & ! [H] :
                                      ( ( v1_funct_1(H)
                                        & v1_funct_2(H,k14_polynom1(A),u1_struct_0(C))
                                        & v2_polynom1(H,k14_polynom1(A),C)
                                        & m2_relset_1(H,k14_polynom1(A),u1_struct_0(C)) )
                                     => ~ ( r2_hidden(H,D)
                                          & r2_polyred(A,B,C,H,F)
                                          & k5_termord(A,B,C,H) = E ) )
                                  & ! [H] :
                                      ( ( v1_funct_1(H)
                                        & v1_funct_2(H,k14_polynom1(A),u1_struct_0(C))
                                        & v2_polynom1(H,k14_polynom1(A),C)
                                        & m2_relset_1(H,k14_polynom1(A),u1_struct_0(C)) )
                                     => ~ ( r2_hidden(H,D)
                                          & r2_polyred(A,B,C,H,G)
                                          & k5_termord(A,B,C,H) = E ) ) )
                               => F = G ) ) ) ) ) ) ) ) )).

fof(t38_groeb_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
            & v1_relat_2(B)
            & v4_relat_2(B)
            & v6_relat_2(B)
            & v8_relat_2(B)
            & v2_bagorder(B,A)
            & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v2_group_1(C)
                & v4_group_1(C)
                & v7_group_1(C)
                & v3_rlvect_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v7_vectsp_1(C)
                & v9_vectsp_1(C)
                & ~ v10_vectsp_1(C)
                & l3_vectsp_1(C) )
             => ! [D] :
                  ( ( ~ v1_xboole_0(D)
                    & v1_ideal_1(D,k30_polynom1(A,C))
                    & v2_ideal_1(D,k30_polynom1(A,C))
                    & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
                 => ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
                     => ! [F] :
                          ( ( v1_funct_1(F)
                            & v1_funct_2(F,k14_polynom1(A),u1_struct_0(C))
                            & v2_polynom1(F,k14_polynom1(A),C)
                            & m2_relset_1(F,k14_polynom1(A),u1_struct_0(C)) )
                         => ! [G] :
                              ( ( v1_funct_1(G)
                                & v1_funct_2(G,k14_polynom1(A),u1_struct_0(C))
                                & v2_polynom1(G,k14_polynom1(A),C)
                                & v1_polynom7(G,A,C)
                                & m2_relset_1(G,k14_polynom1(A),u1_struct_0(C)) )
                             => ( ( r2_hidden(F,E)
                                  & r2_hidden(G,E)
                                  & r3_polynom1(A,k3_termord(A,B,C,G),k3_termord(A,B,C,F))
                                  & r2_groeb_1(A,B,C,E,D) )
                               => ( F = G
                                  | r2_groeb_1(A,B,C,k6_subset_1(u1_struct_0(k30_polynom1(A,C)),E,k1_groeb_1(A,C,F)),D) ) ) ) ) ) ) ) ) ) )).

fof(t39_groeb_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
            & v1_relat_2(B)
            & v4_relat_2(B)
            & v6_relat_2(B)
            & v8_relat_2(B)
            & v2_bagorder(B,A)
            & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v2_group_1(C)
                & v4_group_1(C)
                & v7_group_1(C)
                & v3_rlvect_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v7_vectsp_1(C)
                & v9_vectsp_1(C)
                & ~ v10_vectsp_1(C)
                & l3_vectsp_1(C) )
             => ! [D] :
                  ( ( ~ v1_xboole_0(D)
                    & v1_ideal_1(D,k30_polynom1(A,C))
                    & v2_ideal_1(D,k30_polynom1(A,C))
                    & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
                 => ~ ( D != k1_groeb_1(A,C,k26_polynom1(A,C))
                      & ! [E] :
                          ( ( v1_finset_1(E)
                            & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
                         => ~ ( r2_groeb_1(A,B,C,E,D)
                              & r4_groeb_1(A,B,C,E) ) ) ) ) ) ) ) )).

fof(t40_groeb_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
            & v1_relat_2(B)
            & v4_relat_2(B)
            & v6_relat_2(B)
            & v8_relat_2(B)
            & v2_bagorder(B,A)
            & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v2_group_1(C)
                & v4_group_1(C)
                & v7_group_1(C)
                & v3_rlvect_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v7_vectsp_1(C)
                & v9_vectsp_1(C)
                & ~ v10_vectsp_1(C)
                & l3_vectsp_1(C) )
             => ! [D] :
                  ( ( ~ v1_xboole_0(D)
                    & v1_ideal_1(D,k30_polynom1(A,C))
                    & v2_ideal_1(D,k30_polynom1(A,C))
                    & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
                 => ! [E] :
                      ( ( ~ v1_xboole_0(E)
                        & v1_finset_1(E)
                        & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
                     => ! [F] :
                          ( ( ~ v1_xboole_0(F)
                            & v1_finset_1(F)
                            & m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
                         => ( ( r2_groeb_1(A,B,C,E,D)
                              & r4_groeb_1(A,B,C,E)
                              & r2_groeb_1(A,B,C,F,D)
                              & r4_groeb_1(A,B,C,F) )
                           => E = F ) ) ) ) ) ) ) )).

fof(dt_k1_groeb_1,axiom,(
    ! [A,B,C] :
      ( ( v3_ordinal1(A)
        & v2_group_1(B)
        & v4_rlvect_1(B)
        & v5_rlvect_1(B)
        & v6_rlvect_1(B)
        & v7_vectsp_1(B)
        & ~ v3_realset2(B)
        & l3_vectsp_1(B)
        & v1_funct_1(C)
        & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
        & v2_polynom1(C,k14_polynom1(A),B)
        & m1_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
     => ( ~ v1_xboole_0(k1_groeb_1(A,B,C))
        & v1_finset_1(k1_groeb_1(A,B,C))
        & m1_subset_1(k1_groeb_1(A,B,C),k1_zfmisc_1(u1_struct_0(k30_polynom1(A,B)))) ) ) )).

fof(redefinition_k1_groeb_1,axiom,(
    ! [A,B,C] :
      ( ( v3_ordinal1(A)
        & v2_group_1(B)
        & v4_rlvect_1(B)
        & v5_rlvect_1(B)
        & v6_rlvect_1(B)
        & v7_vectsp_1(B)
        & ~ v3_realset2(B)
        & l3_vectsp_1(B)
        & v1_funct_1(C)
        & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
        & v2_polynom1(C,k14_polynom1(A),B)
        & m1_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
     => k1_groeb_1(A,B,C) = k1_tarski(C) ) )).

fof(dt_k2_groeb_1,axiom,(
    ! [A,B,C,D] :
      ( ( v3_ordinal1(A)
        & v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
        & v1_relat_2(B)
        & v4_relat_2(B)
        & v6_relat_2(B)
        & v8_relat_2(B)
        & m1_relset_1(B,k14_polynom1(A),k14_polynom1(A))
        & ~ v3_struct_0(C)
        & v2_group_1(C)
        & v4_rlvect_1(C)
        & v5_rlvect_1(C)
        & v6_rlvect_1(C)
        & v7_vectsp_1(C)
        & ~ v3_realset2(C)
        & l3_vectsp_1(C)
        & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
     => m1_subset_1(k2_groeb_1(A,B,C,D),k1_zfmisc_1(k14_polynom1(A))) ) )).

fof(dt_k3_groeb_1,axiom,(
    ! [A,B] :
      ( ( v3_ordinal1(A)
        & m1_subset_1(B,k1_zfmisc_1(k14_polynom1(A))) )
     => m1_subset_1(k3_groeb_1(A,B),k1_zfmisc_1(k14_polynom1(A))) ) )).

fof(dt_k4_groeb_1,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ( v1_partfun1(k4_groeb_1(A),k14_polynom1(A),k14_polynom1(A))
        & v1_relat_2(k4_groeb_1(A))
        & v4_relat_2(k4_groeb_1(A))
        & v8_relat_2(k4_groeb_1(A))
        & m2_relset_1(k4_groeb_1(A),k14_polynom1(A),k14_polynom1(A)) ) ) )).

fof(d1_groeb_1,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
            & v1_relat_2(B)
            & v4_relat_2(B)
            & v6_relat_2(B)
            & v8_relat_2(B)
            & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v2_group_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v7_vectsp_1(C)
                & ~ v3_realset2(C)
                & l3_vectsp_1(C) )
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
                 => k2_groeb_1(A,B,C,D) = a_4_0_groeb_1(A,B,C,D) ) ) ) ) )).

fof(d2_groeb_1,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k14_polynom1(A)))
         => k3_groeb_1(A,B) = a_2_0_groeb_1(A,B) ) ) )).

fof(fraenkel_a_4_0_groeb_1,axiom,(
    ! [A,B,C,D,E] :
      ( ( v3_ordinal1(B)
        & v1_partfun1(C,k14_polynom1(B),k14_polynom1(B))
        & v1_relat_2(C)
        & v4_relat_2(C)
        & v6_relat_2(C)
        & v8_relat_2(C)
        & m2_relset_1(C,k14_polynom1(B),k14_polynom1(B))
        & ~ v3_struct_0(D)
        & v2_group_1(D)
        & v4_rlvect_1(D)
        & v5_rlvect_1(D)
        & v6_rlvect_1(D)
        & v7_vectsp_1(D)
        & ~ v3_realset2(D)
        & l3_vectsp_1(D)
        & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k30_polynom1(B,D)))) )
     => ( r2_hidden(A,a_4_0_groeb_1(B,C,D,E))
      <=> ? [F] :
            ( v1_funct_1(F)
            & v1_funct_2(F,k14_polynom1(B),u1_struct_0(D))
            & v2_polynom1(F,k14_polynom1(B),D)
            & m2_relset_1(F,k14_polynom1(B),u1_struct_0(D))
            & A = k3_termord(B,C,D,F)
            & r2_hidden(F,E)
            & F != k26_polynom1(B,D) ) ) ) )).

fof(fraenkel_a_2_0_groeb_1,axiom,(
    ! [A,B,C] :
      ( ( v3_ordinal1(B)
        & m1_subset_1(C,k1_zfmisc_1(k14_polynom1(B))) )
     => ( r2_hidden(A,a_2_0_groeb_1(B,C))
      <=> ? [D] :
            ( m1_polynom1(D,B,k14_polynom1(B))
            & A = D
            & ? [E] :
                ( v7_seqm_3(E)
                & v1_polynom1(E)
                & m1_pboole(E,B)
                & r2_hidden(E,C)
                & r3_polynom1(B,E,D) ) ) ) ) )).
%------------------------------------------------------------------------------