
theorem Th1:
  for S, T being non empty reflexive RelStr, f being Function of S, T,
  P being lower Subset of T st f is monotone holds f"P is lower
proof
  let S, T be non empty reflexive RelStr;
  let f be Function of S, T;
  let P be lower Subset of T;
  assume
A1: f is monotone;
  for x, y being Element of S st x in f"P & y <= x holds y in f"P
  proof
    let x, y be Element of S;
    assume that
A2: x in f"P and
A3: y <= x;
A4: f.y <= f.x by A1,A3;
    reconsider fy = f.y, fx = f.x as Element of T;
    fx in P by A2,FUNCT_2:38;
    then fy in P by A4,WAYBEL_0:def 19;
    hence thesis by FUNCT_2:38;
  end;
  hence thesis;
end;
