
theorem
  for n being Element of NAT, T being connected admissible TermOrder of
  n, L being add-associative right_complementable right_zeroed commutative
  associative well-unital distributive Abelian almost_left_invertible non
degenerated non empty doubleLoopStr, G being non empty Subset of Polynom-Ring
(n,L) holds G is_Groebner_basis_wrt T iff for f being non-zero Polynomial of n,
  L st f in G-Ideal holds f has_a_Standard_Representation_of G,T
proof
  let n be Element of NAT, T be connected admissible TermOrder of n, L be
  add-associative right_complementable right_zeroed commutative associative
  well-unital distributive Abelian almost_left_invertible non degenerated non
  empty doubleLoopStr, P be non empty Subset of Polynom-Ring(n,L);
A1: now
    assume for f being non-zero Polynomial of n,L st f in P-Ideal holds f
    has_a_Standard_Representation_of P,T;
    then for f being non-zero Polynomial of n,L st f in P-Ideal holds f
    is_top_reducible_wrt P,T by Th39;
    then
    for b being bag of n st b in HT(P-Ideal,T) ex b9 being bag of n st b9
    in HT(P,T) & b9 divides b by GROEB_1:18;
    then HT(P-Ideal,T) c= multiples(HT(P,T)) by GROEB_1:19;
    then PolyRedRel(P,T) is locally-confluent by GROEB_1:20;
    hence P is_Groebner_basis_wrt T by GROEB_1:def 3;
  end;
A2: 0_(n,L) = 0.Polynom-Ring(n,L) by POLYNOM1:def 11;
  now
    assume P is_Groebner_basis_wrt T;
    then PolyRedRel(P,T) is locally-confluent by GROEB_1:def 3;
    hence for f being non-zero Polynomial of n,L st f in P-Ideal holds f
    has_a_Standard_Representation_of P,T by A2,Th38,GROEB_1:15;
  end;
  hence thesis by A1;
end;
