
theorem
  for S,T being non empty TopPoset for X being non empty filtered Subset of S
  for f being monotone Function of S,T
  for Y being non empty filtered Subset of T st Y = f.:X
  holds Lim (Y opp+id) c= Lim (f*(X opp+id))
proof
  let S,T be non empty TopPoset;
  let X be non empty filtered Subset of S;
  let f be monotone Function of S,T;
  let Y be non empty filtered Subset of T;
  assume Y = f.:X;
  then f*(X opp+id) is subnet of Y opp+id by Th30;
  hence thesis by YELLOW_6:32;
end;
