
theorem
  for n being Ordinal, T being connected admissible TermOrder of n, L
  being add-associative right_complementable right_zeroed well-unital
  distributive domRing-like non trivial doubleLoopStr, p being non-zero
Polynomial of n,L, q being Polynomial of n,L, m being non-zero Monomial of n,L
  holds HT(p,T) in Support q implies HT(m*'p,T) in Support(m*'q)
proof
  let n be Ordinal, T be connected admissible TermOrder of n, L be
  add-associative right_complementable right_zeroed well-unital distributive
domRing-like non trivial doubleLoopStr, p be non-zero Polynomial of n,L, q be
  Polynomial of n,L, m be non-zero Monomial of n,L;
  set a = coefficient(m), b = term(m);
  assume HT(p,T) in Support q;
  then
A1: q.(HT(p,T)) <> 0.L by POLYNOM1:def 4;
A2: HC(m,T) <> 0.L;
  then reconsider a as non zero Element of L by TERMORD:23;
  m = Monom(a,b) by POLYNOM7:11;
  then m *' p = a * (b *' p) by Th22;
  then HT(m*'p,T) = HT(b*'p,T) by Th21
    .= b + HT(p,T) by Th15;
  then
A3: (m*'q).(HT(m*'p,T)) = m.term(m) * q.(HT(p,T)) by Th7;
  m.(HT(m,T)) <> 0.L by A2,TERMORD:def 7;
  then m.term(m) <> 0.L by POLYNOM7:def 5;
  then (m*'q).(HT(m*'p,T)) <> 0.L by A1,A3,VECTSP_2:def 1;
  hence thesis by POLYNOM1:def 4;
end;
