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

theorem
  for X be non empty set,F be Filter of X,B be non empty Subset of F holds
  F is_coarser_than B iff B is filter_basis
  proof
    let X be non empty set,F be Filter of X,B be non empty Subset of F;
    hereby
      assume
A1:   F is_coarser_than B;
      now
        let f be Element of F;
        consider b be set such that
A2:     b in B and
A3:     b c= f by A1;
        reconsider b1=b as Element of B by A2;
        b1 is Element of B & b1 c= f by A3;
        hence ex b be Element of B st b c= f;
      end;
      hence B is filter_basis;
    end;
    assume
A4: B is filter_basis;
    for f be set st f in F holds ex b be set st b in B & b c= f
    proof
      let f be set;
      assume f in F;
      then consider b be Element of B such that
A5:   b c= f by A4;
      thus thesis by A5;
    end;
    hence thesis;
  end;
