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

theorem Th30Thm70:
  for A be Element of BoolePoset X holds
  {Y where Y is Subset of X : A c= Y} is Filter of BoolePoset X
  proof
    let A be Element of BoolePoset X;
    {Y where Y is Subset of X: A c= Y} = uparrow A by WAYBEL15:2;
    hence thesis;
  end;
