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

theorem
  for X be set,B be Subset-Family of X st {} in B holds <.B.]=bool X
  proof
    let X be set,B be Subset-Family of X;
    assume
A1: {} in B;
    thus <.B.] c= bool X;
    let x be object;
    assume x in bool X;
    then reconsider x0=x as Subset of X;
    {} is Element of B & {} c= x0 by A1;
    hence x in <.B.] by def3;
  end;
