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

theorem Th03:
  for X be non empty set,F be Filter of X holds F is basis of F
  proof
    let X be non empty set,
    F be Filter of X;
    thus F is filter_basis non empty Subset of F
    proof
      F is non empty & F c= F;
      then reconsider F0=F as non empty Subset of F;
      F0 is filter_basis;
      hence thesis;
    end;
  end;
