
theorem Th4:   :: 1.2(iv), p. 39
  for L being lower-bounded antisymmetric reflexive non empty RelStr
  for x being Element of L holds Bottom L << x
proof
  let L be lower-bounded antisymmetric reflexive non empty RelStr;
  let x be Element of L;
  let D be non empty directed Subset of L;
  assume x <= sup D;
  set d = the Element of D;
  reconsider d as Element of L;
  take d;
  thus thesis by YELLOW_0:44;
end;
