
theorem Th5:
  for L being complete antisymmetric non empty RelStr, X being
  lower Subset of L st sup X in X holds X = downarrow sup X
proof
  let L be complete antisymmetric non empty RelStr, X be lower Subset of L
  such that
A1: sup X in X;
  X is_<=_than sup X by YELLOW_0:32;
  hence X c= downarrow sup X by YELLOW_2:1;
  thus thesis by A1,WAYBEL11:6;
end;
