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

theorem
  for X be set, S be Subset-Family of X holds
  <.S.]={x where x is Subset of X: ex b be Element of S st b c= x}
  proof
    let X be set,S be Subset-Family of X;
    set SX={x where x is Subset of X: ex b be Element of S st b c= x};
    hereby
      let x be object;
      assume
A1:   x in <.S.];
      then reconsider x1=x as Subset of X;
      ex b be Element of S st b c= x1 by A1,def3;
      hence x in SX;
    end;
      let x be object;
      assume x in SX;
      then consider x0 be Subset of X such that
A2:   x=x0 and
A3:   ex b be Element of S st b c= x0;
      reconsider x1=x as Subset of X by A2;
      thus x in <.S.] by A2,A3,def3;
  end;
