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

theorem Th15:
  for L being continuous complete non empty Poset, f being
  Function of FixedUltraFilters X, the carrier of L, h being CLHomomorphism of
  InclPoset Filt BoolePoset X, L st h | FixedUltraFilters X = f holds h = f
  -extension_to_hom
proof
  let L be continuous complete non empty Poset, f be Function of
  FixedUltraFilters X, the carrier of L, h be CLHomomorphism of InclPoset Filt
  BoolePoset X, L;
  assume
A1: h | FixedUltraFilters X = f;
  set F = f-extension_to_hom;
  set cL = the carrier of L;
  set BP = BoolePoset X;
  set IP = InclPoset Filt BP;
  set cIP = the carrier of IP;
A2: InclPoset Filt BP = RelStr(#Filt BP, RelIncl Filt BP#) by YELLOW_1:def 1;
  reconsider F9 = F as Function of cIP, cL;
  reconsider h9 = h as Function of cIP, cL;
A3: the carrier of BP = the carrier of LattPOSet BooleLatt X by YELLOW_1:def 2
    .= bool X by LATTICE3:def 1;
  now
    set FUF = FixedUltraFilters X;
    let Fi be Element of cIP;
    Fi in Filt BP by A2;
    then consider FF being Filter of BP such that
A4: Fi = FF;
    set Xsf = {"/\"({f.(uparrow x) : ex z st x = {z} & z in Y}, L) where Y is
    Subset of X : Y in FF };
    set Xs = {"/\"({uparrow x : ex z st x = {z} & z in Y}, IP) where Y is
    Subset of X : Y in FF };
A5: Xs c= cIP
    proof
      let p be object;
      assume p in Xs;
      then
      ex YY being Subset of X st p = "/\"({uparrow x : ex z st x = {z} & z
      in YY}, IP) & YY in FF;
      hence thesis;
    end;
    now
      consider YY being object such that
A6:   YY in FF by XBOOLE_0:def 1;
    reconsider YY as set by TARSKI:1;
      "/\"({uparrow x : ex z st x = {z} & z in YY}, IP) in Xs by A3,A6;
      hence Xs is non empty;
    end;
    then reconsider Xs as non empty Subset of IP by A5;
A7: ex_sup_of Xs, IP by YELLOW_0:17;
A8: Xs is directed
    proof
      let a, b be Element of IP;
      assume that
A9:   a in Xs and
A10:  b in Xs;
      consider Yb being Subset of X such that
A11:  b = "/\"({uparrow x : ex z st x = {z} & z in Yb}, IP) and
A12:  Yb in FF by A10;
      reconsider Yb9 = Yb as Element of FF by A12;
      set Xsb = {uparrow x : ex z st x = {z} & z in Yb};
      consider Ya being Subset of X such that
A13:  a = "/\"({uparrow x : ex z st x = {z} & z in Ya}, IP) and
A14:  Ya in FF by A9;
      reconsider Ya9 = Ya as Element of FF by A14;
      set Xsa = {uparrow x : ex z st x = {z} & z in Ya};
      per cases;
      suppose
A15:    Xsa is empty;
        take a;
        thus a in Xs by A9;
        thus a <= a;
        "/\"(Xsa, IP) = Top IP by A15;
        hence b <= a by A13,YELLOW_0:45;
      end;
      suppose
A16:    Xsb is empty;
        take b;
        thus b in Xs by A10;
        "/\"(Xsb, IP) = Top IP by A16;
        hence a <= b by A11,YELLOW_0:45;
        thus b <= b;
      end;
      suppose
A17:    Xsa is non empty & Xsb is non empty;
        Xsb c= cIP
        proof
          let r be object;
          assume r in Xsb;
          then
          ex xz being Element of BP st r = uparrow xz & ex z st xz = {z} &
          z in Yb;
          hence thesis by A2;
        end;
        then reconsider Xsb as non empty Subset of IP by A17;
        Xsa c= cIP
        proof
          let r be object;
          assume r in Xsa;
          then
          ex xz being Element of BP st r = uparrow xz & ex z st xz = {z} &
          z in Ya;
          hence thesis by A2;
        end;
        then reconsider Xsa as non empty Subset of IP by A17;
A18:    "/\"(Xsb, IP) = meet Xsb by WAYBEL16:10;
        consider Yab being Element of BP such that
A19:    Yab in FF and
A20:    Yab <= Ya9 and
A21:    Yab <= Yb9 by WAYBEL_0:def 2;
        reconsider Yab as Element of FF by A19;
        set c = "/\"({uparrow x : ex z st x = {z} & z in Yab}, IP);
        set Xsc = {uparrow x : ex z st x = {z} & z in Yab};
A22:    "/\"(Xsa, IP) = meet Xsa by WAYBEL16:10;
        thus thesis
        proof
          per cases;
          suppose
A23:        Xsc is empty;
            take c;
            thus c in Xs by A3;
A24:        "/\"(Xsc, IP) = Top IP by A23;
            hence a <= c by YELLOW_0:45;
            thus b <= c by A24,YELLOW_0:45;
          end;
          suppose
A25:        Xsc is non empty;
            Xsc c= cIP
            proof
              let r be object;
              assume r in Xsc;
              then
              ex xz being Element of BP st r = uparrow xz & ex z st xz = {
              z} & z in Yab;
              hence thesis by A2;
            end;
            then reconsider Xsc as non empty Subset of IP by A25;
            take c;
            thus c in Xs by A3;
A26:        "/\"(Xsc, IP) = meet Xsc by WAYBEL16:10;
            a c= c
            proof
              let d be object;
              Xsc c= Xsa
              proof
                let r be object;
                assume r in Xsc;
                then
A27:            ex xz being Element of BP st r = uparrow xz & ex z st xz =
                {z} & z in Yab;
                Yab c= Ya by A20,YELLOW_1:2;
                hence thesis by A27;
              end;
              then
A28:          meet Xsa c= meet Xsc by SETFAM_1:6;
              assume d in a;
              hence thesis by A13,A22,A26,A28;
            end;
            hence a <= c by YELLOW_1:3;
            b c= c
            proof
              let d be object;
              Xsc c= Xsb
              proof
                let r be object;
                assume r in Xsc;
                then
A29:            ex xz being Element of BP st r = uparrow xz & ex z st xz =
                {z} & z in Yab;
                Yab c= Yb by A21,YELLOW_1:2;
                hence thesis by A29;
              end;
              then
A30:          meet Xsb c= meet Xsc by SETFAM_1:6;
              assume d in b;
              hence thesis by A11,A18,A26,A30;
            end;
            hence b <= c by YELLOW_1:3;
          end;
        end;
      end;
    end;
A31: h is infs-preserving by WAYBEL16:def 1;
    now
      let s be object;
      hereby
        assume s in h.:Xs;
        then consider t being object such that
        t in the carrier of IP and
A32:    t in Xs and
A33:    s = h.t by FUNCT_2:64;
        consider Y being Subset of X such that
A34:    t = "/\"({uparrow x : ex z st x = {z} & z in Y}, IP) & Y in FF by A32;
        set Xsi = {uparrow x : ex z st x = {z} & z in Y};
        Xsi c= cIP
        proof
          let r be object;
          assume r in Xsi;
          then
          ex xz being Element of BP st r = uparrow xz & ex z being Element
          of X st xz = {z} & z in Y;
          hence thesis by A2;
        end;
        then reconsider Xsi as Subset of IP;
        set Xsif = {f.(uparrow x) : ex z st x = {z} & z in Y};
        now
          let u be object;
          hereby
            assume u in h.:Xsi;
            then consider v being object such that
            v in the carrier of IP and
A35:        v in Xsi and
A36:        u = h.v by FUNCT_2:64;
A37:        ex x st v = uparrow x & ex z st x = {z} & z in Y by A35;
            then v in FUF;
            then h.v = f.v by A1,FUNCT_1:49;
            hence u in Xsif by A36,A37;
          end;
          assume u in Xsif;
          then consider x such that
A38:      u = f.(uparrow x) and
A39:      ex z st x = {z} & z in Y;
          uparrow x in FUF by A39;
          then
A40:      h.(uparrow x) = f.(uparrow x) by A1,FUNCT_1:49;
          uparrow x in Xsi by A39;
          hence u in h.:Xsi by A38,A40,FUNCT_2:35;
        end;
        then
A41:    h.:Xsi = Xsif by TARSKI:2;
        h preserves_inf_of Xsi & ex_inf_of Xsi, IP by A31,YELLOW_0:17;
        then inf (h.:Xsi) = h.inf Xsi;
        hence s in Xsf by A33,A34,A41;
      end;
      assume s in Xsf;
      then consider Y being Subset of X such that
A42:  s = "/\"({f.(uparrow x) : ex z st x = {z} & z in Y}, L) and
A43:  Y in FF;
      set Xsi = {uparrow x : ex z st x = {z} & z in Y};
      Xsi c= cIP
      proof
        let r be object;
        assume r in Xsi;
        then ex xz being Element of BP st r = uparrow xz & ex z being Element
        of X st xz = {z} & z in Y;
        hence thesis by A2;
      end;
      then reconsider Xsi as Subset of IP;
      set Xsif = {f.(uparrow x) : ex z st x = {z} & z in Y};
      h preserves_inf_of Xsi & ex_inf_of Xsi, IP by A31,YELLOW_0:17;
      then
A44:  inf (h.:Xsi) = h.inf Xsi;
      now
        let u be object;
        hereby
          assume u in h.:Xsi;
          then consider v being object such that
          v in the carrier of IP and
A45:      v in Xsi and
A46:      u = h.v by FUNCT_2:64;
A47:      ex x st v = uparrow x & ex z st x = {z} & z in Y by A45;
          then v in FUF;
          then h.v = f.v by A1,FUNCT_1:49;
          hence u in Xsif by A46,A47;
        end;
        assume u in Xsif;
        then consider x such that
A48:    u = f.(uparrow x) and
A49:    ex z st x = {z} & z in Y;
        uparrow x in FUF by A49;
        then
A50:    h.(uparrow x) = f.(uparrow x) by A1,FUNCT_1:49;
        uparrow x in Xsi by A49;
        hence u in h.:Xsi by A48,A50,FUNCT_2:35;
      end;
      then
A51:  h.:Xsi = Xsif by TARSKI:2;
      inf Xsi in Xs by A43;
      hence s in h.:Xs by A42,A44,A51,FUNCT_2:35;
    end;
    then
A52: h.:Xs = Xsf by TARSKI:2;
A53: FF = "\/"({"/\"({uparrow x : ex z st x = {z} & z in Y}, IP) where Y is
    Subset of X : Y in FF }, IP) by Th11;
    h is directed-sups-preserving by WAYBEL16:def 1;
    then h preserves_sup_of Xs by A8;
    hence h9.Fi = sup (h.:Xs) by A4,A53,A7
      .= F9.Fi by A4,A52,Def3;
  end;
  hence thesis by FUNCT_2:63;
end;
