
theorem
  for L be antisymmetric with_infima RelStr for S be Subset of L holds S
  is meet-closed iff for x,y be Element of L st x in S & y in S holds inf {x,y}
  in S
proof
  let L be antisymmetric with_infima RelStr;
  let S be Subset of L;
  thus S is meet-closed implies for x,y be Element of L st x in S & y in S
  holds inf {x,y} in S
  by YELLOW_0:21,Th15;
  assume for x,y be Element of L st x in S & y in S holds inf {x,y} in S;
  then
  for x,y be Element of L st x in S & y in S & ex_inf_of {x,y},L holds inf
  {x,y} in S;
  hence thesis by Th15;
end;
