
theorem Th18:
  for L be antisymmetric with_suprema RelStr for S be Subset of L
holds S is join-closed iff for x,y be Element of L st x in S & y in S holds sup
  {x,y} in S
proof
  let L be antisymmetric with_suprema RelStr;
  let S be Subset of L;
  thus S is join-closed implies for x,y be Element of L st x in S & y in S
  holds sup {x,y} in S
  by YELLOW_0:20,Th16;
  assume for x,y be Element of L st x in S & y in S holds sup {x,y} in S;
  then
  for x,y be Element of L st x in S & y in S & ex_sup_of {x,y},L holds sup
  {x,y} in S;
  hence thesis by Th16;
end;
