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

theorem Th20:
  for X be non empty set,
  L be non empty transitive reflexive RelStr,
  f be Function of [#]L,X,
  B be filter_base of X st [#]L is directed holds
  B is_coarser_than f.:#(Tails L) iff
  for b be Element of B ex i be Element of L st
  for j be Element of L st i <=j holds f.j in b
  proof
    let X be non empty set, L be non empty transitive reflexive RelStr,
    f be Function of [#]L,X,
    B be filter_base of X;
    assume
A1: [#]L is directed;
    hereby
      assume
A2:   B is_coarser_than f.:#(Tails L);
      hereby
        let b be Element of B;
        consider x0 be set such  that
A3:     x0 in f.:#(Tails L) and
A4:     x0 c= b by A2;
        reconsider x1=x0 as Subset of X by A4,XBOOLE_1:1;
        consider j0 be Element of L such that
A5: for i be Element of L st i>=j0 holds f.i in x1 by A1,A3,Th17;
        for i be Element of L st i >=j0 holds f.i in b by A4,A5;
        hence ex i be Element of L st
        for j be Element of L st i <=j holds f.j in b;
      end;
    end;
    assume
A6: for b be Element of B ex i be Element of L st
    for j be Element of L st i <=j holds f.j in b;
    now
      let Y be set;
      assume
A7:   Y in B;
      then consider i0 be Element of L such that
A8:   for j be Element of L st i0 <= j holds f.j in Y by A6;
      reconsider b=Y as Subset of X by A7;
      consider b0 be Element of Tails L such that
A9:   f.:b0 c= b by A1,A8,Th18;
      f.:b0 in f.:#(Tails L) by FUNCT_2:def 10;
      hence ex X be set st X in f.:#(Tails L) & X c= Y by A9;
    end;
    hence B is_coarser_than f.:#(Tails L);
  end;
