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

theorem Th06:
  for X be non empty set,F be Filter of X,B be basis of F holds F=<.#B.]
  proof
    let X be non empty set,F be Filter of X, B be basis of F;
    hereby
      let x be object;
      assume x in F;
      then reconsider x0=x as Element of F;
      consider b0 be Element of B such that
A1:   b0 c= x0 by def2;
      reconsider x0 as Subset of X;
      thus x in <.#B.] by A1,def3;
    end;
    let x be object;
    assume
A2: x in <.#B.];
    then reconsider x0=x as Subset of X;
    reconsider B2=#B as Subset-Family of X;
    consider b0 be Element of B such that
A3: b0 c= x0 by A2,def3;
    thus x in F by A3,CARD_FIL:def 1;
  end;
