
theorem Th1: :: 1.2(i), p. 39
  for L being non empty reflexive antisymmetric RelStr
  for x,y being Element of L st x << y holds x <= y
proof
  let L be non empty reflexive antisymmetric RelStr;
  let x,y be Element of L such that
A1: for D being non empty directed Subset of L st y <= sup D
  ex d being Element of L st d in D & x <= d;
A2: {y} is directed by WAYBEL_0:5;
  sup {y} = y by YELLOW_0:39;
  then ex d being Element of L st ( d in {y})&( x <= d) by A1,A2;
  hence thesis by TARSKI:def 1;
end;
