
theorem Th18:
  for L being non empty RelStr, x,y being Element of L
  holds y in uparrow x iff x <= y
proof
  let L be non empty RelStr, x,y be Element of L;
A1: uparrow x = {z where z is Element of L:
  ex v being Element of L st z >= v & v in {x}} by Th15;
  then y in uparrow x iff ex z being Element of L st y = z &
  ex v being Element of L st z >= v & v in {x};
  hence y in uparrow x implies y >= x by TARSKI:def 1;
  x in {x} by TARSKI:def 1;
  hence thesis by A1;
end;
