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

theorem Th14:
  for L be non empty transitive reflexive RelStr st [#]L is directed
  holds <.Tails L.] is Filter of [#]L
  proof
    let L be non empty transitive reflexive RelStr;
    assume
A1: [#]L is directed;
    Tails L is non empty Subset-Family of L &
    Tails L is with_non-empty_elements &
    Tails L is quasi_basis
    proof
A2:   Tails L is with_non-empty_elements
      proof
        assume not Tails L is with_non-empty_elements;
        then consider i0 be Element of L such that
A3:     {}=uparrow i0;
        thus contradiction by A3;
      end;
      (for x,y be Element of Tails L
      ex z be Element of Tails L st z c= x/\y)
      proof
        let x,y be Element of Tails L;
        x in Tails L;
        then consider lx be Element of L such that
A4:     x=uparrow lx;
        y in Tails L;
        then consider ly be Element of L such that
A5:     y=uparrow ly;
        consider lz be Element of L such that
        lz in the carrier of L and
A6:     lx <= lz and
A7:     ly <= lz by A1;
        set z=uparrow lz;
        z in Tails L;
        then reconsider z as Element of Tails L;
        take z;
        uparrow lz c= uparrow lx & uparrow lz c= uparrow ly
        by A6,A7,WAYBEL_0:22;
        hence thesis by A4,A5,XBOOLE_1:19;
      end;
      hence thesis by A2;
    end;
    hence thesis by Th08;
  end;
