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