
theorem Th19:
  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) holds
Upper_Support(p,T,i) \/ Lower_Support(p,T,i) = Support p & Upper_Support(p,T,i)
  /\ Lower_Support(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;
  set M = Upper_Support(p,T,i) /\ (Support(p)\Upper_Support(p,T,i));
  assume i <= card(Support p);
  then
A1: Upper_Support(p,T,i) c= Support p by Def2;
  thus Upper_Support(p,T,i) \/ Lower_Support(p,T,i) = Upper_Support(p,T,i) \/
  Support p by XBOOLE_1:39
    .= Support p by A1,XBOOLE_1:12;
  now
    set x = the Element of M;
    assume M <> {};
    then x in Upper_Support(p,T,i) & x in Support(p)\Upper_Support(p,T,i) by
XBOOLE_0:def 4;
    hence contradiction by XBOOLE_0:def 5;
  end;
  hence thesis;
end;
