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,B1,B2 be basis of F holds
  #B1, #B2 are_equivalent_generators
  proof
    let X be non empty set,F be Filter of X,B1,B2 be basis of F;
    hereby
      let b1 be Element of #B1;
      b1 in #B1;
      then b1 in F;
      then b1 in <.#B2.] by Th06;
      hence ex b2 be Element of #B2 st b2 c= b1 by def3;
    end;
    let b2 be Element of #B2;
    b2 in #B2;
    then b2 in F;
    then b2 in <.#B1.] by Th06;
    hence ex b1 be Element of #B1 st b1 c= b2 by def3;
end;
