
theorem
  for A, B, C being non empty RelStr, f being Function of B, C, g, h
  being Function of A, B st g <= h & f is monotone holds f * g <= f * h
proof
  let A, B, C be non empty RelStr, f be Function of B, C, g, h be Function of
  A, B such that
A1: g <= h and
A2: for x, y being Element of B st x <= y holds f.x <= f.y;
  for x being Element of A holds (f*g).x <= (f*h).x
  proof
    let x be Element of A;
A3: (f*g).x = f.(g.x) & (f*h).x = f.(h.x) by FUNCT_2:15;
    g.x <= h.x by A1,YELLOW_2:9;
    hence thesis by A2,A3;
  end;
  hence thesis by YELLOW_2:9;
end;
