reserve L for complete Scott TopLattice,
  x for Element of L,
  X, Y for Subset of L,
  V, W for Element of InclPoset sigma L,
  VV for Subset of InclPoset sigma L;

theorem Th25:
  for X being filtered Subset of L st VV = {(downarrow x)` : x in
  X} holds VV is directed
proof
  let V be filtered Subset of L;
  set F = {downarrow u where u is Element of L : u in V};
  F c= bool the carrier of L
  proof
    let x be object;
    assume x in F;
    then ex u being Element of L st x = downarrow u & u in V;
    hence thesis;
  end;
  then reconsider F as Subset-Family of L;
  reconsider F as Subset-Family of L;
  assume
A1: VV = {(downarrow x)` : x in V};
  VV c= bool the carrier of L
  proof
    let x be object;
    assume x in VV;
    then ex u being Element of L st x = (downarrow u)` & u in V by A1;
    hence thesis;
  end;
  then reconsider VV as Subset-Family of L;
  reconsider VV as Subset-Family of L;
  now
    let x be object;
    hereby
      assume x in VV;
      then consider u being Element of L such that
A2:   x = (downarrow u)` and
A3:   u in V by A1;
      downarrow u in F by A3;
      hence x in COMPLEMENT F by A2,YELLOW_8:5;
    end;
    assume
A4: x in COMPLEMENT F;
    then reconsider X = x as Subset of L;
    X` in F by A4,SETFAM_1:def 7;
    then consider u being Element of L such that
A5: X` = downarrow u and
A6: u in V;
    X = (downarrow u)` by A5;
    hence x in VV by A1,A6;
  end;
  then VV = COMPLEMENT F by TARSKI:2;
  hence thesis by Lm2;
end;
