
theorem Th30:
  for n being Ordinal, T being connected TermOrder of n, L being
  add-associative right_zeroed right_complementable non empty addLoopStr, p
being Polynomial of n,L, i being Element of NAT st 1 <= i & i <= card(Support p
  ) holds HT(p,T) in Support Up(p,T,i)
proof
  let n be Ordinal, T be connected TermOrder of n, L be add-associative
right_zeroed right_complementable non empty addLoopStr, p be Polynomial of n,
  L, i be Element of NAT;
  assume that
A1: 1 <= i and
A2: i <= card(Support p);
  Support p <> {} by A1,A2;
  then
A3: HT(p,T) in Support p by TERMORD:def 6;
  set u = Up(p,T,i);
  set x = the Element of Support u;
A4: Support u = Upper_Support(p,T,i) by A2,Lm3;
  then card (Support u) <> 0 by A1,A2,Def2;
  then
A5: Support u <> {};
  then
A6: x in Support u;
  then reconsider x as Element of Bags n;
  Support u c= Support p by A2,A4,Def2;
  then x <= HT(p,T),T by A6,TERMORD:def 6;
  hence thesis by A2,A4,A5,A3,Def2;
end;
