
theorem Th24: :: according to definition 5.15, p. 194
  for n being Ordinal, T being connected TermOrder of n, L being
non trivial ZeroStr, p being non-zero Polynomial of n,L holds PosetMax(Support(
  p,T)) = HT(p,T)
proof
  let n be Ordinal, T be connected TermOrder of n, L be non trivial ZeroStr,
  p be non-zero Polynomial of n,L;
  set htp = HT(p,T), R = RelStr(#Bags n,T#), sp = Support(p,T);
  p <> 0_(n,L) by POLYNOM7:def 1;
  then Support(p) <> {} by POLYNOM7:1;
  then
A1: htp in Support(p) by TERMORD:def 6;
  now
    assume
    ex y being set st y in sp & y <> htp & [htp,y] in the InternalRel of R;
    then consider y being set such that
A2: y in sp and
A3: y <> htp and
A4: [htp,y] in the InternalRel of R;
    y is Element of Bags n by A2;
    then reconsider y9 = y as bag of n;
    y9 <= htp,T & htp <= y9,T by A2,A4,TERMORD:def 2,def 6;
    hence contradiction by A3,TERMORD:7;
  end;
  then htp is_maximal_wrt Support(p,T),the InternalRel of R by A1,
WAYBEL_4:def 23;
  hence thesis by A1,BAGORDER:def 13;
end;
