
theorem
  for L being non empty reflexive antisymmetric RelStr
  for x,y being Element of L st uparrow x = uparrow y holds x = y
proof
  let L be non empty reflexive antisymmetric RelStr;
  let x,y be Element of L;
  reconsider x9 = x, y9 = y as Element of L;
A1: x9 <= x9;
A2: y9 <= y9;
  assume
A3: uparrow x = uparrow y;
  then
A4: y in uparrow x by A2,Th18;
A5: x in uparrow y by A1,A3,Th18;
A6: x9 <= y9 by A4,Th18;
  x9 >= y9 by A5,Th18;
  hence thesis by A6,ORDERS_2:2;
end;
