
theorem Th19:
  for n being Ordinal, T being connected TermOrder of n, L being
non trivial ZeroStr, p being Polynomial of n,L, b being bag of n st b <> HT(p,T
  ) holds HM(p,T).b = 0.L
proof
  let n being Ordinal, O be connected TermOrder of n, L be non trivial ZeroStr
  , p be Polynomial of n,L, b being bag of n;
  assume
A1: b <> HT(p,O);
  per cases by POLYNOM7:6;
  suppose
    Support HM(p,O) = {};
    then HM(p,O) = 0_(n,L) by POLYNOM7:1;
    hence thesis by POLYNOM1:22;
  end;
  suppose
    ex b being bag of n st Support HM(p,O) = {b};
    then consider b1 being bag of n such that
A2: Support HM(p,O) = {b1};
A3: b is Element of Bags n by PRE_POLY:def 12;
    now
      per cases;
      case
        HC(p,O) <> 0.L;
        then HT(p,O) in {b1} by A2,Lm9;
        then b1 <> b by A1,TARSKI:def 1;
        then not b in {b1} by TARSKI:def 1;
        hence thesis by A2,A3,POLYNOM1:def 4;
      end;
      case
        HC(p,O) = 0.L;
        then Support(HM(p,O)) = {} by Lm10;
        then HM(p,O) = 0_(n,L) by POLYNOM7:1;
        hence thesis by POLYNOM1:22;
      end;
    end;
    hence thesis;
  end;
end;
