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

theorem Th11:
  F = "\/"({"/\"({uparrow x : ex z st x = {z} & z in Y}, InclPoset
Filt BoolePoset X) where Y is Subset of X : Y in F}, InclPoset Filt BoolePoset
  X)
proof
  set BP = BoolePoset X;
  set IP = InclPoset Filt BP;
  set cIP = the carrier of IP;
  set Xs = {"/\"({uparrow x : ex z st x = {z} & z in Y }, IP ) where Y is
  Subset of X : Y in F};
  set RS = "\/"(Xs, IP);
A1: InclPoset Filt BP = RelStr(#Filt BP, RelIncl Filt BP#) by YELLOW_1:def 1;
  F in Filt BP;
  then reconsider F9 = F as Element of IP by A1;
A2: 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 F;
    hence thesis;
  end;
A3: the carrier of BP = the carrier of LattPOSet BooleLatt X by YELLOW_1:def 2
    .= bool X by LATTICE3:def 1;
  now
    consider YY being object such that
A4: YY in F 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,A4;
    hence Xs is non empty;
  end;
  then reconsider Xs9 = Xs as non empty Subset of IP by A2;
A5: ex_sup_of Xs9, IP by YELLOW_0:17;
  F c= RS
  proof
    let p be object;
    assume
A6: p in F;
    then reconsider Y = p as Element of F;
    set Xsi = {uparrow x where x is Element of BP : ex z being Element of X st
    x = {z} & z in Y};
A7: "/\"(Xsi, IP) in Xs by A3;
    per cases;
    suppose
A8:   Xsi is empty;
A9:   p in the carrier of BP by A6;
      Xs9 is_<=_than RS by A5,YELLOW_0:def 9;
      then
A10:  "/\"(Xsi, IP) <= RS by A7;
      "/\"(Xsi, IP) = Top IP by A8
        .= bool X by WAYBEL16:15;
      then bool X c= RS by A10,YELLOW_1:3;
      hence thesis by A3,A9;
    end;
    suppose
A11:  Xsi is non empty;
      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 A1;
      end;
      then reconsider Xsi as non empty Subset of IP by A11;
      for yy being set st yy in Xsi holds Y in yy
      proof
        reconsider Y9 = Y as Element of BP;
        let yy be set;
        assume yy in Xsi;
        then consider r being Element of BP such that
A12:    yy = uparrow r and
A13:    ex z being Element of X st r = {z} & z in Y;
        r c= Y by A13,ZFMISC_1:31;
        then r <= Y9 by YELLOW_1:2;
        hence thesis by A12,WAYBEL_0:18;
      end;
      then "/\"(Xsi, IP) = meet Xsi & Y in meet Xsi by SETFAM_1:def 1
,WAYBEL16:10;
      then
A14:  p in union Xs by A7,TARSKI:def 4;
      union Xs9 c= RS by WAYBEL16:17,YELLOW_0:17;
      hence thesis by A14;
    end;
  end;
  then
A15: F9 <= RS by YELLOW_1:3;
  Xs is_<=_than F9
  proof
    let b be Element of IP;
    assume b in Xs;
    then consider Y being Subset of X such that
A16: b = "/\"({uparrow x : ex z st x = {z} & z in Y}, IP) and
A17: Y in F;
    reconsider Y9 = Y as Element of F by A17;
    set Xsi = {uparrow x : ex z st x = {z} & z in Y};
    per cases;
    suppose
A18:  Y is empty;
      now
        assume Xsi is non empty;
        then consider p being object such that
A19:    p in Xsi by XBOOLE_0:def 1;
        ex x being Element of BP st p = uparrow x & ex z being Element of
        X st x = {z} & z in Y by A19;
        hence contradiction by A18;
      end;
      then
A20:  "/\"(Xsi, IP) = Top IP .= bool X by WAYBEL16:15;
      Bottom BP = {} by YELLOW_1:18;
      then uparrow Bottom BP c= F by A17,A18,WAYBEL11:42;
      then bool X c= F by A3,WAYBEL14:10;
      hence b <= F9 by A3,A16,A20,XBOOLE_0:def 10;
    end;
    suppose
A21:  Y is non empty;
A22:  now
        consider z being object such that
A23:    z in Y by A21,XBOOLE_0:def 1;
        reconsider z as Element of X by A23;
        reconsider x = {z} as Element of BP by A3,A23,ZFMISC_1:31;
        uparrow x in Xsi by A23;
        hence Xsi is non empty;
      end;
      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 A1;
      end;
      then reconsider Xsi as non empty Subset of IP by A22;
A24:  "/\"(Xsi, IP) = meet Xsi by WAYBEL16:10;
      b c= F9
      proof
        let yy be object;
        b in Filt BP by A1;
        then consider bf being Filter of BP such that
A25:    b = bf;
        assume
A26:    yy in b;
        then reconsider yy9 = yy as Element of bf by A25;
        reconsider yy9 as Element of BP;
        Y c= yy9
        proof
          let zz be object;
          assume
A27:      zz in Y;
          then reconsider z = zz as Element of X;
          reconsider xz = {z} as Element of BP by A3,A27,ZFMISC_1:31;
          uparrow xz in Xsi by A27;
          then yy in uparrow xz by A16,A24,A26,SETFAM_1:def 1;
          then xz <= yy9 by WAYBEL_0:18;
          then {z} c= yy9 by YELLOW_1:2;
          hence thesis by ZFMISC_1:31;
        end;
        then Y9 <= yy9 by YELLOW_1:2;
        then uparrow Y9 c= F9 & yy in uparrow Y9 by WAYBEL11:42,WAYBEL_0:18;
        hence thesis;
      end;
      hence b <= F9 by YELLOW_1:3;
    end;
  end;
  then RS <= F9 by A5,YELLOW_0:def 9;
  hence thesis by A15,YELLOW_0:def 3;
end;
