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

theorem
  for X be non empty set, A be non empty Subset of X holds for
  F be Filter of A, B be basis of F holds
  (incl A).:( #B ) is filter_base of X &
  <.((incl A).:( #B )).] is Filter of X &
  <.((incl A).:( #B )).]= { M where M is Subset of X: (incl A)"(M) in F}
  by Th12;
