
theorem Th28:
  for 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 st i <= card(Support p) for b being bag
of n st b in Support p holds (b in Support Up(p,T,i) or b in Support Low(p,T,i)
  ) & not(b in Support Up(p,T,i) /\ Support Low(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
A1: i <= card(Support p);
  let b being bag of n;
  assume
A2: b in Support p;
  Support p = Upper_Support(p,T,i) \/ Lower_Support(p,T,i) by A1,Th19
    .= Support Up(p,T,i) \/ Lower_Support(p,T,i) by A1,Lm3
    .= Support Up(p,T,i) \/ Support Low(p,T,i) by A1,Lm3;
  hence b in Support Up(p,T,i) or b in Support Low(p,T,i) by A2,XBOOLE_0:def 3;
  Support Up(p,T,i) = Upper_Support(p,T,i) & Support Low(p,T,i) =
  Lower_Support(p,T,i) by A1,Lm3;
  hence thesis by A1,Th19;
end;
