
theorem Th20:
  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,
  b9 being bag of n st b in Upper_Support(p,T,i) & b9 in Lower_Support(p,T,i)
  holds b9 < b,T
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,b9 being bag of n;
  assume that
A2: b in Upper_Support(p,T,i) and
A3: b9 in Lower_Support(p,T,i);
A4: Lower_Support(p,T,i) c= Support p by XBOOLE_1:36;
  now
    assume b <= b9,T;
    then b9 in Upper_Support(p,T,i) by A1,A2,A3,A4,Def2;
    then b9 in Upper_Support(p,T,i) /\ Lower_Support(p,T,i) by A3,
XBOOLE_0:def 4;
    hence contradiction by A1,Th19;
  end;
  hence thesis by TERMORD:5;
end;
