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

theorem Th10:
  for X be non empty set, B be filter_base of X holds B is basis of <.B.)
  proof
    let X be non empty set,
    B be filter_base of X;
    reconsider B2=<.B.) as Filter of X;
    B is filter_basis non empty Subset of B2
    proof
      B is non empty Subset of B2
      proof
        B c= B2 by def3;
        hence thesis;
      end;
      then reconsider B3=B as non empty Subset of B2;
      B3 is filter_basis by def3;
      hence thesis;
    end;
    hence thesis;
  end;
