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

theorem Th09:
  for X be non empty set, F be Filter of X, B be basis of F holds
  B is filter_base of X
  proof
    let X be non empty set, F be Filter of X, B be basis of F;
    for b1,b2 be Element of B holds ex b be Element of B st b c= b1/\b2
    proof
      let b1,b2 be Element of B;
      b1/\b2 in F by CARD_FIL:def 1;
      then consider b0 be Element of B such that
A1:   b0 c= b1/\b2 by def2;
      thus thesis by A1;
    end;
    then
A2: #B is quasi_basis;
    B is with_non-empty_elements by CARD_FIL:def 1;
    hence thesis by A2;
  end;
