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

theorem Th17:
  for X be non empty set, L be non empty transitive reflexive RelStr,
  f be Function of [#]L,X, x be Subset of X st [#]L is directed
  & x in f.:#(Tails L) holds
  ex j be Element of L st for i be Element of L st i >= j holds f.i in x
  proof
    let X be non empty set,L be non empty transitive reflexive RelStr,
    f be Function of [#]L,X, x be Subset of X;
    assume that [#]L is directed and
A2: x in f.:#(Tails L);
    reconsider x0=x as Subset of X;
    consider b0 be Subset of [#]L such that
A3: b0 in #(Tails L) and
A4: x0=f.:b0 by A2,FUNCT_2:def 10;
    consider i be Element of L such that
A5: b0=uparrow i by A3;
    now
      let j be Element of L;
      assume j >= i;
      then C1: i <= j & i in {i} by TARSKI:def 1;
      j in [#]L;
      then j in uparrow i & j in dom f by C1,WAYBEL_0:def 16,FUNCT_2:def 1;
      hence f.j in x by A4,A5,FUNCT_1:def 6;
    end;
    hence thesis;
  end;
