
theorem Th3:
  for L be upper-bounded antisymmetric non empty RelStr holds ((the
  carrier of L) \ { Top L }), { Top L } form_upper_lower_partition_of L
proof
  let L be upper-bounded antisymmetric non empty RelStr;
A1: for a,b be Element of L st a in ((the carrier of L) \ { Top L }) & b in
  { Top L } holds a < b
  proof
    let a,b be Element of L;
    assume that
A2: a in ((the carrier of L) \ { Top L }) and
A3: b in { Top L };
    not a in { Top L } by A2,XBOOLE_0:def 5;
    then
A4: a <> Top L by TARSKI:def 1;
A5: a <= Top L by YELLOW_0:45;
    b = Top L by A3,TARSKI:def 1;
    hence thesis by A4,A5,ORDERS_2:def 6;
  end;
  ((the carrier of L) \ { Top L }) \/ { Top L } = the carrier of L by
XBOOLE_1:45;
  hence thesis by A1;
end;
