reserve x, X, Y for set;

theorem
  for S, T being non empty RelStr for f being Function of S, T for X
  being directed Subset of S holds f is monotone implies f.:X is directed
proof
  let S, T be non empty RelStr;
  let f be Function of S, T;
  let X be directed Subset of S;
  set Y = f.:X;
  assume
A1: f is monotone;
  for y1, y2 being Element of T st y1 in Y & y2 in Y ex z being Element of
  T st z in Y & y1 <= z & y2 <= z
  proof
    let y1, y2 be Element of T;
    assume that
A2: y1 in Y and
A3: y2 in Y;
    consider x1 being object such that
    x1 in dom f and
A4: x1 in X and
A5: y1 = f.x1 by A2,FUNCT_1:def 6;
    consider x2 being object such that
    x2 in dom f and
A6: x2 in X and
A7: y2 = f.x2 by A3,FUNCT_1:def 6;
    reconsider x1, x2 as Element of S by A4,A6;
    consider z being Element of S such that
A8: z in X and
A9: x1 <= z & x2 <= z by A4,A6,WAYBEL_0:def 1;
    take f.z;
    thus f.z in Y by A8,FUNCT_2:35;
    thus thesis by A1,A5,A7,A9;
  end;
  hence thesis;
end;
