
theorem Th35:
  for n being Ordinal, T being connected admissible TermOrder of n
  , L being add-associative right_complementable right_zeroed non trivial
  addLoopStr, p being non-zero Polynomial of n,L holds Red(p,T) < p,T
proof
  let n be Ordinal, T be connected admissible TermOrder of n, L be
  add-associative right_complementable right_zeroed non trivial addLoopStr, p
  be non-zero Polynomial of n,L;
  (Red(p,T)).(HT(p,T)) = 0.L by TERMORD:39;
  then
A1: not HT(p,T) in Support(Red(p,T)) by POLYNOM1:def 4;
  p <> 0_(n,L) by POLYNOM7:def 1;
  then Support p <> {} by POLYNOM7:1;
  then
A2: HT(p,T) in Support p by TERMORD:def 6;
  Red(p,T) < HM(p,T),T by Th33;
  then
A3: Red(p,T) <= HM(p,T),T;
  HM(p,T) <= p,T by Th34;
  then Red(p,T) <= p,T by A3,Th27;
  hence thesis by A2,A1;
end;
