
theorem Th26:
  for n being Ordinal, T being connected TermOrder of n, L being
  non trivial ZeroStr, p being Polynomial of n,L holds HT(HM(p,T),T) = HT(p,T)
proof
  let n be Ordinal, O be connected TermOrder of n, L be non trivial ZeroStr, p
  be Polynomial of n,L;
  per cases;
  suppose
    HC(p,O) is non zero;
    then reconsider a = HC(p,O) as non zero Element of L;
    thus HT(HM(p,O),O) = term(Monom(a,HT(p,O))) by Lm11
      .= HT(p,O) by POLYNOM7:10;
  end;
  suppose
A1: not HC(p,O) is non zero;
    now
      per cases;
      case
        not p is non-zero;
        then p = 0_(n,L) by POLYNOM7:def 1;
        then Support p = {} by POLYNOM7:1;
        then HT(p,O) = EmptyBag n by Def6
          .= term(Monom(0.L,HT(p,O))) by POLYNOM7:8
          .= term(HM(p,O)) by A1,STRUCT_0:def 12;
        hence thesis by Lm11;
      end;
      case
        p is non-zero;
        then reconsider p as non-zero Polynomial of n,L;
        HC(p,O) is non zero;
        hence thesis by A1;
      end;
    end;
    hence thesis;
  end;
end;
