
theorem Th2:
  for S, T being RelStr, K, L being non empty RelStr for f being
Function of S, T, g being Function of K, L st the RelStr of S = the RelStr of K
& the RelStr of T = the RelStr of L & f = g & f is antitone holds g is antitone
proof
  let S, T be RelStr, K, L be non empty RelStr, f be Function of S, T, g be
  Function of K, L such that
A1: the RelStr of S = the RelStr of K and
A2: the RelStr of T = the RelStr of L and
A3: f = g and
A4: f is antitone;
  reconsider S1 = S, T1 = T as non empty RelStr by A1,A2;
  let x, y be Element of K such that
A5: x <= y;
  reconsider x1 = x, y1 = y as Element of S1 by A1;
  reconsider f1 = f as Function of S1, T1;
  x1 <= y1 by A1,A5;
  then f1.x1 >= f1.y1 by A4;
  hence thesis by A2,A3;
end;
