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

theorem Th19:
  for X be non empty set,
  L be non empty transitive reflexive RelStr,
  f be Function of [#]L,X,
  F be Filter of X, B be basis of F st [#]L is directed holds
  F is_filter-coarser_than filter_image(f,Tails_Filter(L)) iff
  B is_coarser_than f.:#(Tails L)
  proof
    let X be non empty set,
    L be non empty transitive reflexive RelStr,
    f be Function of [#]L,X,
    F be Filter of X, B be basis of F;
    assume [#]L is directed;
    then f.:#(Tails L) is basis of filter_image(f,Tails_Filter(L))
    by Th16;
    hence thesis by Th11;
  end;
