reserve X for set,
  F for Filter of BoolePoset X,
  x for Element of BoolePoset X ,
  z for Element of X;

theorem Th13:
  for L being continuous complete non empty Poset, f being
  Function of FixedUltraFilters X, the carrier of L holds (f-extension_to_hom).
  Top (InclPoset Filt BoolePoset X) = Top L
proof
  let L be continuous complete non empty Poset, f be Function of
  FixedUltraFilters X, the carrier of L;
  set IP = InclPoset Filt BoolePoset X;
  set F = f-extension_to_hom;
  reconsider T = Top IP as Element of IP;
  reconsider E = {} as Subset of X by XBOOLE_1:2;
  set Z = {"/\"({f.(uparrow x) : ex z st x = {z} & z in Y }, L) where Y is
  Subset of X : Y in T};
A1: {f.(uparrow x) : ex z st x = {z} & z in E} = {}
  proof
    assume not thesis;
    then consider r being object such that
A2: r in {f.(uparrow x) : ex z st x = {z} & z in E} by XBOOLE_0:def 1;
    ex x st r = f.(uparrow x) & ex z st x = {z} & z in E by A2;
    hence contradiction;
  end;
A3: F.T = "\/"({"/\"({f.(uparrow x) : ex z st x = {z} & z in Y }, L) where Y
  is Subset of X : Y in T}, L) by Def3;
  T = bool X by WAYBEL16:15;
  then
A4: "/\"({f.(uparrow x) : ex z st x = {z} & z in E}, L) in Z;
  Z is_<=_than "\/"(Z, L) by YELLOW_0:32;
  then Top L <= "\/"(Z, L) by A4,A1;
  hence thesis by A3,WAYBEL15:3;
end;
