reserve x, X, Y for set;

theorem Th16:
  for L being with_suprema Poset for f being Function of L, L
  holds f is directed-sups-preserving implies f is monotone
proof
  let L be with_suprema Poset;
  let f be Function of L, L;
  assume
A1: f is directed-sups-preserving;
  let x, y be Element of L such that
A2: x <= y;
A3: y = y"\/"x by A2,YELLOW_0:24;
  for a, b being Element of L st a in {x, y} & b in {x, y} ex z being
  Element of L st z in {x, y} & a <= z & b <= z
  proof
    let a, b be Element of L such that
A4: a in {x, y} & b in {x, y};
    take y;
    thus y in {x, y} by TARSKI:def 2;
    thus thesis by A2,A4,TARSKI:def 2;
  end;
  then {x, y} is directed non empty;
  then
A5: f preserves_sup_of {x, y} by A1;
A6: dom f = the carrier of L by FUNCT_2:def 1;
  y <= y;
  then
A7: {x, y} is_<=_than y by A2,YELLOW_0:8;
  for b being Element of L st {x, y} is_<=_than b holds y <= b by YELLOW_0:8;
  then ex_sup_of {x, y},L by A7,YELLOW_0:30;
  then sup(f.:{x, y}) = f.sup{x, y} by A5
    .= f.y by A3,YELLOW_0:41;
  then
A8: f.y = sup{f.x, f.y} by A6,FUNCT_1:60
    .= f.y"\/"f.x by YELLOW_0:41;
  let afx, bfy be Element of L;
  assume afx = f.x & bfy = f.y;
  hence afx <= bfy by A8,YELLOW_0:22;
end;
