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

theorem
  for X,Y be non empty set, f be Function of X,Y,
  B1,B2 be filter_base of X st B1 is_coarser_than B2
  holds <.B1.) is_filter-coarser_than <.B2.)
  proof
    let X,Y be non empty set, f be Function of X,Y,
    B1,B2 be filter_base of X;
    assume
A1: B1 is_coarser_than B2;
    B1 is basis of <.B1.) & B2 is basis of <.B2.) by Th10;
    hence thesis by A1,Th11;
  end;
