
theorem Th31: :: lemma 5.17 (i), p. 195
  for n being Ordinal, T being admissible connected TermOrder of n
  , L being add-associative right_complementable right_zeroed well-unital
  distributive domRing-like non trivial doubleLoopStr, p,q being non-zero
  Polynomial of n,L holds HT(p*'q,T) = HT(p,T) + HT(q,T)
proof
  let n be Ordinal, O be admissible connected TermOrder of n, L be
  add-associative right_complementable right_zeroed well-unital distributive
  domRing-like non trivial doubleLoopStr, p,q be non-zero Polynomial of n,L;
A1: HT(p,O)+HT(q,O) is Element of Bags n by PRE_POLY:def 12;
  HT(p,O) + HT(q,O) in Support(p*'q) by Th29;
  then HT(p,O) + HT(q,O) <= HT(p*'q,O),O by Def6;
  then
A2: [HT(p,O) + HT(q,O),HT(p*'q,O)] in O;
  Support p*'q <> {} by Th29;
  then
A3: HT(p*'q,O) in Support(p*'q) by Def6;
  Support(p*'q) c= {s + t where s,t is Element of Bags n : s in Support p
  & t in Support q} by Th30;
  then
  HT(p*'q,O) in {s + t where s,t is Element of Bags n : s in Support p & t
  in Support q} by A3;
  then consider s,t being Element of Bags n such that
A4: HT(p*'q,O) = s + t and
A5: s in Support p and
A6: t in Support q;
  s <= HT(p,O),O by A5,Def6;
  then [s,HT(p,O)] in O;
  then
A7: [s + t,HT(p,O) + t] in O by BAGORDER:def 5;
  t <= HT(q,O),O by A6,Def6;
  then [t,HT(q,O)] in O;
  then
A8: [t + HT(p,O), HT(p,O)+HT(q,O)] in O by BAGORDER:def 5;
  s + t is Element of Bags n & HT(p,O) + t is Element of Bags n by
PRE_POLY:def 12;
  then [s + t,HT(p,O)+HT(q,O)] in O by A1,A7,A8,ORDERS_1:5;
  hence thesis by A2,A4,A1,ORDERS_1:4;
end;
