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

theorem
  for X be non empty set, FB be filter_base of X,F be Filter of X st
  FB c= F holds <.FB.) is_coarser_than F
  proof
    let X be non empty set,FB be filter_base of X,F be Filter of X;
    assume
A1: FB c= F;
    now let x be object;
      assume
A2:   x in <.FB.);
      then reconsider x1=x as Subset of X;
      consider b be Element of FB such that
A4:   b c= x1 by A2,def3;
A5:   b in F by A1;
      reconsider x2=x1 as Subset of X;
      consider c be Element of F such that
A6:   c c= x2 by A4,A5;
      thus x in F by A6,CARD_FIL:def 1;
    end;
    hence thesis;
  end;
