
theorem
  for L being upper-bounded with_suprema antisymmetric RelStr for X
  being non empty Subset of L holds X "\/" {Top L} = {Top L}
proof
  let L be upper-bounded with_suprema antisymmetric RelStr, X be non empty
  Subset of L;
  thus X "\/" {Top L} c= {Top L} by Th12;
  let q be object;
  assume q in {Top L};
  then
A1: X "\/" {Top L} = {Top L "\/" y where y is Element of L: y in X} & q =
  Top L by TARSKI:def 1,YELLOW_4:15;
  consider y being object such that
A2: y in X by XBOOLE_0:def 1;
  reconsider y as Element of X by A2;
  Top L "\/" y = Top L by WAYBEL_1:4;
  hence thesis by A1;
end;
