
theorem Th3:
  for S, T being reflexive antisymmetric non empty RelStr,
  f being Function of S, T st
  f is directed-sups-preserving holds f is monotone
proof
  let S, T be reflexive antisymmetric non empty RelStr, f be Function of S, T;
  assume
A1: f is directed-sups-preserving;
  let x, y be Element of S such that
A2: x <= y;
  y <= y;
  then
A3: {x, y} is_<=_than y by A2,YELLOW_0:8;
  A4: for
 b being Element of S st {x, y} is_<=_than b holds y <= b by YELLOW_0:8;
  then
A5: y = sup {x, y} by A3,YELLOW_0:30;
A6: ex_sup_of {x, y},S by A3,A4,YELLOW_0:30;
  for a, b being Element of S st a in {x, y} & b in {x, y}
  ex z being Element of S st z in {x, y} & a <= z & b <= z
  proof
    let a, b be Element of S such that
A7: a in {x, y} and
A8: b in {x, y};
    take y;
    thus y in {x, y} by TARSKI:def 2;
    thus thesis by A2,A7,A8,TARSKI:def 2;
  end;
  then {x, y} is directed non empty;
  then
A9: f preserves_sup_of {x, y} by A1;
  then
A10: sup(f.:{x, y}) = f.y by A5,A6;
A11: dom f = the carrier of S by FUNCT_2:def 1;
  then
A12: f.y = sup{f.x, f.y} by A10,FUNCT_1:60;
  f.:{x, y} = {f.x, f.y} by A11,FUNCT_1:60;
  then ex_sup_of {f.x, f.y}, T by A6,A9;
  then {f.x, f.y} is_<=_than f.y by A12,YELLOW_0:def 9;
  hence f.x <= f.y by YELLOW_0:8;
end;
