reserve x, X, Y for set;

theorem
  for L being with_infima Poset for f being Function of L, L holds f is
  filtered-infs-preserving implies f is monotone
proof
  let L be with_infima Poset;
  let f be Function of L, L;
  assume
A1: f is filtered-infs-preserving;
  let x, y be Element of L such that
A2: x <= y;
A3: x = x"/\"y by A2,YELLOW_0:25;
  for c, d being Element of L st c in {x, y} & d in {x, y} ex z being
  Element of L st z in {x, y} & z <= c & z <= d
  proof
    let c, d be Element of L such that
A4: c in {x, y} & d in {x, y};
    take x;
    thus x in {x, y} by TARSKI:def 2;
    thus thesis by A2,A4,TARSKI:def 2;
  end;
  then {x, y} is filtered non empty;
  then
A5: f preserves_inf_of {x, y} by A1;
A6: dom f = the carrier of L by FUNCT_2:def 1;
  x <= x;
  then
A7: x is_<=_than {x, y} by A2,YELLOW_0:8;
  for c being Element of L st c is_<=_than {x, y} holds c <= x by YELLOW_0:8;
  then ex_inf_of {x, y},L by A7,YELLOW_0:31;
  then inf(f.:{x, y}) = f.inf{x, y} by A5
    .= f.x by A3,YELLOW_0:40;
  then
A8: f.x = inf{f.x, f.y} by A6,FUNCT_1:60
    .= f.x"/\"f.y by YELLOW_0:40;
  let a, b be Element of L;
  assume a = f.x & b = f.y;
  hence a <= b by A8,YELLOW_0:23;
end;
