
theorem :: theorem 5.35 (viii) ==> (ix), p. 206
  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, P
  being Subset of Polynom-Ring(n,L) holds (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) implies HT(P
  -Ideal,T) c= multiples(HT(P,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 be Subset of
  Polynom-Ring(n,L);
  assume
A1: 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;
    let u be object;
    assume
A2: u in HT(P-Ideal,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;
