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

theorem Th35:
  for X be non empty set,F be non empty Subset of BooleLatt X holds
  F is with_non-empty_elements & F is Filter of BooleLatt X iff
  F is Filter of X
  proof
    let X be non empty set,
    F be non empty Subset of BooleLatt X;
    hereby
      assume that
A1:   F is with_non-empty_elements and
A2:   F is Filter of BooleLatt X;
A3:   F is non empty Subset-Family of X by LATTICE3:def 1;
      for Y1,Y2 be Subset of X holds
      (Y1 in F & Y2 in F implies Y1/\Y2 in F) &
      (Y1 in F & Y1 c= Y2 implies Y2 in F) by A2,Th32;
      hence F is Filter of X by A1,A3,CARD_FIL:def 1;
    end;
    assume F is Filter of X;
    then F is with_non-empty_elements &
    for Y1,Y2 be Subset of X holds
    (Y1 in F & Y2 in F implies Y1/\Y2 in F) &
    (Y1 in F & Y1 c= Y2 implies Y2 in F) by CARD_FIL:def 1;
    hence F is with_non-empty_elements &
    F is Filter of BooleLatt X  by Th32;
  end;
