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,FX be Filter of X st FX={X} holds
  FX is_coarser_than F
  proof
    let X be non empty set,F be Filter of X,FX be Filter of X;
    assume
A1: FX={X};
    X in F by CARD_FIL:5;
    then {X} c= F by TARSKI:def 1;
    hence thesis by A1;
  end;
