reserve
  X for non empty set,
  FX for Filter of X,
  SFX for Subset-Family of X;

theorem Th16:
  for X be non empty set,
  L be non empty transitive reflexive RelStr,
  f be Function of [#]L,X st [#]L is directed
  holds
  f.:#(Tails L) is basis of filter_image(f,Tails_Filter(L))
  proof
    let X be non empty set,
    L be non empty transitive reflexive RelStr,
    f be Function of [#]L,X such that A1: [#]L is directed;
    reconsider SL = Tails(L) as basis of Tails_Filter(L)
    by A1,Th15;
    f.:#SL is basis of filter_image(f,Tails_Filter(L)) by Th13bThmBA2;
    hence thesis;
  end;
