
theorem Th23:
  for n being Ordinal, T being connected TermOrder of n, L being
  add-associative right_zeroed right_complementable non trivial addLoopStr, p
being non-zero Polynomial of n,L, i being Element of NAT st 1 <= i & i <= card(
  Support p) holds HT(p,T) in Upper_Support(p,T,i)
proof
  let n be Ordinal, T be connected TermOrder of n, L be add-associative
  right_zeroed right_complementable non trivial addLoopStr, p be non-zero
  Polynomial of n,L, i be Element of NAT;
  assume that
A1: 1 <= i and
A2: i <= card(Support p);
  p <> 0_(n,L) by POLYNOM7:def 1;
  then Support p <> {} by POLYNOM7:1;
  then
A3: HT(p,T) in Support p by TERMORD:def 6;
  set u = Upper_Support(p,T,i);
  set x = the Element of u;
A4: u <> {} by A1,A2,Def2,CARD_1:27;
  then
A5: x in u;
  then reconsider x9 = x as Element of Bags n;
  u c= Support p by A2,Def2;
  then x9 <= HT(p,T),T by A5,TERMORD:def 6;
  hence thesis by A2,A4,A3,Def2;
end;
