
theorem Th9:
  for n being Ordinal, T being connected admissible TermOrder of n,
  L being add-associative right_complementable right_zeroed commutative
  associative well-unital distributive almost_left_invertible non degenerated
  non empty doubleLoopStr, f,p being non-zero Polynomial of n,L holds f
  is_reducible_wrt p,T implies HT(p,T) <= HT(f,T),T
proof
  let n being Ordinal, T being connected admissible TermOrder of n, L being
  add-associative right_complementable right_zeroed commutative associative
  well-unital distributive almost_left_invertible non degenerated non empty
  doubleLoopStr, f,p being non-zero Polynomial of n,L;
  assume f is_reducible_wrt p,T;
  then consider b being bag of n such that
A1: b in Support f & HT(p,T) divides b by POLYRED:36;
  b <= HT(f,T),T & HT(p,T) <= b,T by A1,TERMORD:10,def 6;
  hence thesis by TERMORD:8;
end;
