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

theorem Th18:
  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
  & (ex j be Element of L st for i be Element of L st i >= j holds f.i in x)
  holds ex b be Element of Tails L st f.:b c= 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
A1: ex j be Element of L st for i be Element of L st i >=j holds f.i in x;
    consider j0 be Element of L such that
A2: for i be Element of L st i >=j0 holds f.i in x by A1;
    set b0=uparrow {j0};
    b0=uparrow j0;
    then
A3: b0 in #(Tails L);
    now
      let t be object;
      assume t in f.:b0;
      then consider x0 be object such that
A4:   x0 in dom f and
A5:   x0 in uparrow j0 and
A6:   t=f.x0 by FUNCT_1:def 6;
      reconsider x1=x0 as Element of L by A4;
      consider y1 be Element of L such that
A7:   y1<= x1 and
A8:   y1 in {j0} by A5,WAYBEL_0:def 16;
      y1=j0 by A8,TARSKI:def 1;
      hence t in x by A2,A6,A7;
    end;
    then f.:b0 c= x;
    hence thesis by A3;
  end;
