
theorem Th29:
  for n being Ordinal, T being admissible connected TermOrder of n
  , L being add-associative right_complementable left_zeroed right_zeroed
  well-unital distributive domRing-like non trivial doubleLoopStr, p,q being
  non-zero Polynomial of n,L holds HT(p,T) + HT(q,T) in Support(p*'q)
proof
  let n be Ordinal, O be admissible connected TermOrder of n, L be
  add-associative right_complementable right_zeroed left_zeroed well-unital
  distributive domRing-like non trivial doubleLoopStr, p,q be non-zero
  Polynomial of n,L;
  set b = HT(p,O) + HT(q,O);
  assume
A1: not HT(p,O) + HT(q,O) in Support(p*'q);
  p <> 0_(n,L) by POLYNOM7:def 1;
  then Support p <> {} by POLYNOM7:1;
  then HT(p,O) in Support p by Def6;
  then
A2: p.(HT(p,O)) <> 0.L by POLYNOM1:def 4;
  q <> 0_(n,L) by POLYNOM7:def 1;
  then Support q <> {} by POLYNOM7:1;
  then HT(q,O) in Support q by Def6;
  then
A3: q.(HT(q,O)) <> 0.L by POLYNOM1:def 4;
  b is Element of Bags n by PRE_POLY:def 12;
  then (p*'q).(HT(p,O) + HT(q,O)) = 0.L by A1,POLYNOM1:def 4;
  then p.(HT(p,O)) * q.(HT(q,O)) = 0.L by Lm14;
  hence thesis by A2,A3,VECTSP_2:def 1;
end;
