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

theorem Th15:
  for L be non empty transitive reflexive RelStr
  st [#]L is directed holds
  Tails L is basis of Tails_Filter(L)
  proof
    let L be non empty transitive reflexive RelStr;
    assume [#]L is directed;
    then Tails_Filter(L)=<.Tails L.] by DefL9;
    hence thesis by Th07;
  end;
