
theorem Th28: :: theorem 5.38 (iv) ==> (v), p. 207
  for n being Ordinal, T being connected TermOrder of n, L being
  add-associative right_complementable right_zeroed commutative associative
well-unital distributive almost_left_invertible non trivial doubleLoopStr, G,
I being Subset of Polynom-Ring(n,L) holds (for b being bag of n st b in HT(I,T)
  ex b9 being bag of n st b9 in HT(G,T) & b9 divides b) implies HT(I,T) c=
  multiples(HT(G,T))
proof
  let n be Ordinal, T be connected TermOrder of n, L be add-associative
  right_complementable right_zeroed commutative associative well-unital
  distributive almost_left_invertible non trivial doubleLoopStr, P,I being
  Subset of Polynom-Ring(n,L);
  assume
A1: for b being bag of n st b in HT(I,T) ex b9 being bag of n st b9 in
  HT(P,T) & b9 divides b;
    let u be object;
    assume
A2: u in HT(I,T);
    then reconsider u9 = u as Element of Bags n;
    ex b9 being bag of n st b9 in HT(P,T) & b9 divides u9 by A1,A2;
    hence u in multiples(HT(P,T));
end;
