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

theorem Th11:
  for X be non empty set, F1,F2 be Filter of X,
  B1 be basis of F1,B2 be basis of F2 holds
  F1 is_filter-coarser_than F2 iff B1 is_coarser_than B2
  proof
    let X be non empty set,
    F1,F2 be Filter of X,
    B1 be basis of F1,B2 be basis of F2;
    hereby
      assume F1 is_filter-coarser_than F2;
      then
A1:   F1 c= F2;
      now
        let x be set;
        assume x in B1;
        then x in F2 by A1;
        then ex b be Element of B2 st b c= x by def2;
        hence ex y be set st y in B2 & y c= x;
      end;
      hence B1 is_coarser_than B2;
    end;
    assume
A2: B1 is_coarser_than B2;
    now
      let x be object;
      assume
A3:   x in F1;
      then reconsider x1=x as Subset of X;
      consider b1 be Element of B1 such that
A4:   b1 c= x1 by A3,def2;
      consider b2 be set such that
A5:   b2 in B2 and
A6:   b2 c= b1 by A2;
      reconsider b2 as Element of B2 by A5;
A7:   b2 c= x1 by A4,A6;
      x1 in <.#B2.] by A7,def3;
      hence x in F2 by Th06;
    end;
    then F1 c= F2;
    hence F1 is_filter-coarser_than F2;
  end;
