
theorem Th17:
  for L being non empty RelStr, x,y being Element of L
  holds y in downarrow x iff y <= x
proof
  let L be non empty RelStr, x,y be Element of L;
A1: downarrow x = {z where z is Element of L:
  ex v being Element of L st z <= v & v in {x}} by Th14;
  then y in downarrow 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 downarrow x implies y <= x by TARSKI:def 1;
  x in {x} by TARSKI:def 1;
  hence thesis by A1;
end;
