
theorem Th45:
  for L being satisfying_MC with_infima antisymmetric non empty
  reflexive RelStr holds for x being Element of L, D being non empty directed
  Subset of L st x <= sup D holds x = sup ({x} "/\" D)
proof
  let L be satisfying_MC with_infima antisymmetric non empty reflexive RelStr;
  let x be Element of L, D be non empty directed Subset of L;
  assume x <= sup D;
  hence x = x "/\" sup D by YELLOW_0:25
    .= sup ({x} "/\" D) by Def6;
end;
