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

theorem Th14:
  for L being continuous complete non empty Poset, f being
  Function of FixedUltraFilters X, the carrier of L holds f-extension_to_hom |
  FixedUltraFilters X = f
proof
  let L be continuous complete non empty Poset, f be Function of
  FixedUltraFilters X, the carrier of L;
  set FUF = FixedUltraFilters X;
  set BP = BoolePoset X;
  set IP = InclPoset Filt BP;
A1: InclPoset Filt BP = RelStr(#Filt BP, RelIncl Filt BP#) by YELLOW_1:def 1;
  set F = f-extension_to_hom;
A2: the carrier of BP = the carrier of LattPOSet BooleLatt X by YELLOW_1:def 2
    .= bool X by LATTICE3:def 1;
  now
A3: dom F = the carrier of IP by FUNCT_2:def 1;
    hence FUF = dom (F|FUF) by A1,Th9,RELAT_1:62;
    thus FUF = dom f by FUNCT_2:def 1;
    let xf be object;
    assume
A4: xf in FUF;
    then reconsider FUF9 = FUF as non empty Subset-Family of BoolePoset X;
A5: (F|FUF).xf = F.xf by A4,FUNCT_1:49;
    FUF c= dom F by A1,A3,Th9;
    then reconsider x9 = xf as Element of IP by A4,FUNCT_2:def 1;
    reconsider xf9 = xf as Element of FUF9 by A4;
    set Xs = {"/\"({f.(uparrow x) : ex z st x = {z} & z in Y }, L) where Y is
    Subset of X : Y in x9 };
    reconsider f9 = f as Function of FUF9, the carrier of L;
    f9.xf9 is Element of L;
    then reconsider fxf = f.xf9 as Element of L;
    consider xx being Element of BoolePoset X such that
A6: xf = uparrow xx and
A7: ex y being Element of X st xx = {y} by A4;
A8: Xs is_<=_than fxf
    proof
      let b be Element of L;
      assume b in Xs;
      then consider Y being Subset of X such that
A9:   b = "/\"({f.(uparrow x) : ex z st x = {z} & z in Y}, L) and
A10:  Y in x9;
      set Xsi = {f.(uparrow x) : ex z st x = {z} & z in Y };
      ex_inf_of Xsi, L by YELLOW_0:17;
      then
A11:  Xsi is_>=_than b by A9,YELLOW_0:def 10;
      reconsider Y as Element of BoolePoset X by A6,A10;
      consider y being Element of X such that
A12:  xx = {y} by A7;
      xx <= Y by A6,A10,WAYBEL_0:18;
      then xx c= Y by YELLOW_1:2;
      then y in Y by A12,ZFMISC_1:31;
      then fxf in Xsi by A6,A12;
      hence b <= fxf by A11;
    end;
A13: for a being Element of L st Xs is_<=_than a holds fxf <= a
    proof
      xx <= xx;
      then reconsider Y = xx as Element of x9 by A6,WAYBEL_0:18;
      set Xsi = {f.(uparrow x) : ex z st x = {z} & z in Y };
      consider y being Element of X such that
A14:  xx = {y} by A7;
      now
        let p be object;
        hereby
          assume p in Xsi;
          then consider r being Element of BoolePoset X such that
A15:      p = f.(uparrow r) and
A16:      ex z being Element of X st r = {z} & z in Y;
          xx = r by A14,A16,TARSKI:def 1;
          hence p in {fxf} by A6,A15,TARSKI:def 1;
        end;
        assume p in {fxf};
        then
A17:    p = fxf by TARSKI:def 1;
        y in Y by A14,TARSKI:def 1;
        hence p in Xsi by A6,A14,A17;
      end;
      then Xsi = {fxf} by TARSKI:2;
      then fxf = "/\"(Xsi, L) by YELLOW_0:39;
      then
A18:  fxf in Xs by A2;
      let a be Element of L;
      assume Xs is_<=_than a;
      hence thesis by A18;
    end;
    ex_sup_of Xs, L by YELLOW_0:17;
    then f.xf = "\/"(Xs, L) by A8,A13,YELLOW_0:def 9;
    hence (F|FUF).xf = f.xf by A5,Def3;
  end;
  hence thesis;
end;
