
theorem Th31:
  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 i <= card(Support p) for b
being bag of n st b in Support Low(p,T,i) holds Low(p,T,i).b = p.b & Up(p,T,i).
  b = 0.L
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;
  set l = Lower_Support(p,T,i);
  assume
A1: i <= card(Support p);
  then
A2: l /\ Upper_Support(p,T,i) = {} by Th19;
  let b be bag of n;
  assume
A3: b in Support Low(p,T,i);
  hence Low(p,T,i).b = p.b by Th16;
  b in l by A1,A3,Lm3;
  then not b in Upper_Support(p,T,i) by A2,XBOOLE_0:def 4;
  then
A4: not b in Support(Up(p,T,i)) by A1,Lm3;
  b is Element of Bags n by PRE_POLY:def 12;
  hence thesis by A4,POLYNOM1:def 4;
end;
