
theorem
  for n being Ordinal, T being connected TermOrder of n, L being non
trivial ZeroStr, p being Polynomial of n,L holds Support(HM(p,T)) c= Support(p)
proof
  let n be Ordinal, O be connected TermOrder of n, L be non trivial ZeroStr, p
  be Polynomial of n,L;
    let u be object;
    assume
A1: u in Support(HM(p,O));
    now
      per cases by A1,Lm12;
      case
        u in {};
        hence u in Support p;
      end;
      case
        u in {HT(p,O)};
        then
A2:     u = HT(p,O) by TARSKI:def 1;
        now
          per cases;
          case
            HC(p,O) = 0.L;
            then (HM(p,O)).(HT(p,O)) = 0.L by Lm8;
            hence contradiction by A1,A2,POLYNOM1:def 4;
          end;
          case
            HC(p,O) <> 0.L;
            hence u in Support p by A2,POLYNOM1:def 4;
          end;
        end;
        hence u in Support p;
      end;
    end;
    hence u in Support p;
end;
