
theorem
  for L being non empty reflexive antisymmetric RelStr
  for x,y being Element of L st downarrow x = downarrow 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: downarrow x = downarrow y;
  then
A4: y in downarrow x by A2,Th17;
  x in downarrow y by A1,A3,Th17;
  then
A5: x9 <= y9 by Th17;
  x9 >= y9 by A4,Th17;
  hence thesis by A5,ORDERS_2:2;
end;
