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

theorem
  for X be non empty set,F be Filter of X,B be basis of F,
  S be Subset-Family of X, S1 be Subset of F
  st S=S1 & #B,S are_equivalent_generators holds S1 is basis of F
  proof
    let X be non empty set,F be Filter of X,B be basis of F,
    S be Subset-Family of X,
    S1 be Subset of F
    such that
A1: S=S1 and
A2: #B,S are_equivalent_generators;
    <.#B.]=<.S.] by A2,Th05;
    hence thesis by A1,Th06,Th07;
  end;
