
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 Subset of Polynom-Ring(n,L)
  holds G is_Groebner_basis_wrt T implies (for g1,g2 being Polynomial of n,L st
  g1 in G & g2 in G & not(HT(g1,T),HT(g2,T) are_disjoint) holds PolyRedRel(G,T)
  reduces S-Poly(g1,g2,T),0_(n,L))
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, G be Subset of Polynom-Ring(n,L);
  assume G is_Groebner_basis_wrt T;
  then for g1,g2,h being Polynomial of n,L st g1 in G & g2 in G & h
  is_a_normal_form_of S-Poly(g1,g2,T),PolyRedRel(G,T) holds h = 0_(n,L) by
GROEB_2:23;
  hence thesis by GROEB_2:24;
end;
