reserve T for non empty TopSpace,
  X,Z for Subset of T;
reserve x,y for Element of OpenClosedSet(T);
reserve x,y,X for set;
reserve BL for non trivial B_Lattice,
  a,b,c,p,q for Element of BL,
  UF,F,F0,F1,F2 for Filter of BL;

theorem Th32:
  ultraset BL = union X & X is Subset of StoneR BL implies
  ex Y being Element of Fin X st ultraset BL = union Y
proof
  assume that
A1: ultraset BL = union X and
A2: X is Subset of StoneR BL;
  assume
A3: not thesis;
  consider Y being Element of Fin X such that
A4: Y is non empty by A1,Th27,ZFMISC_1:2;
A5: Y c= X by FINSUB_1:def 5;
  then
A6: Y c= StoneR BL by A2,XBOOLE_1:1;
  set x = the Element of Y;
A7: x in X by A4,A5;
  x in StoneR BL by A4,A6;
  then consider b such that
A8: x =UFilter BL.b by Th23;
  set CY = { a` : UFilter BL.a in X };
  consider c such that
A9: c = b`;
A10: c in CY by A7,A8,A9;
  CY c= the carrier of BL
  proof
    let x be object;
    assume x in CY;
    then ex b st ( x = b`)&( UFilter BL.b in X);
    hence thesis;
  end;
  then reconsider CY as non empty Subset of BL by A10;
  set H = <.CY.);
  for B being non empty Element of Fin the carrier of BL st B c= CY
  holds FinMeet B <> Bottom BL
  proof
    let B be non empty Element of Fin the carrier of BL such that
A11: B c= CY and
A12: FinMeet B = Bottom BL;
A13: B is Subset of BL by FINSUB_1:16;
A14: dom UFilter BL = the carrier of BL by FUNCT_2:def 1;
    UFilter BL.:B c= rng UFilter BL by RELAT_1:111;
    then reconsider D = UFilter BL.:B
    as non empty Subset-Family of ultraset BL by A13,A14,XBOOLE_1:1;
A15: now
      set x = the Element of UFilter BL.Bottom BL;
      assume {}ultraset BL <> UFilter BL.Bottom BL;
      then ex F st F=x & F is being_ultrafilter & Bottom BL in F by Th17;
      hence contradiction by Th29;
    end;
    deffunc U(Subset of ultraset BL,Subset of ultraset BL) = $1 /\ $2;
    consider G being BinOp of bool ultraset BL such that
A16: for x,y being Subset of ultraset BL holds
    G.(x,y)= U(x,y) from BINOP_1:sch 4;
A17: G is commutative
    proof
      let x,y be Subset of ultraset BL;
      G. (x,y) = y /\ x by A16
        .= G. (y,x) by A16;
      hence thesis;
    end;
A18: G is associative
    proof
      let x,y,z be Subset of ultraset BL;
      G. (x,G. (y,z)) = G.(x, y /\ z) by A16
        .= x /\ (y /\ z) by A16
        .= (x /\ y) /\ z by XBOOLE_1:16
        .= G.(x /\ y, z) by A16
        .= G. (G.(x,y),z) by A16;
      hence thesis;
    end;
A19: G is idempotent
    proof
      let x be Subset of ultraset BL;
      G. (x,x) = x /\ x by A16
        .= x;
      hence thesis;
    end;
A20: for x,y being Element of BL holds
    UFilter BL.((the L_meet of BL).(x,y)) = G.(UFilter BL.x,UFilter BL.y)
    proof
      let x,y be Element of BL;
      thus UFilter BL.((the L_meet of BL).(x,y)) = UFilter BL.(x "/\" y)
        .=UFilter BL.x /\ UFilter BL.y by Th20
        .=G.(UFilter BL.x,UFilter BL.y) by A16;
    end;
    reconsider DD = D as Element of Fin bool ultraset BL;
    id BL = id the carrier of BL;
    then
A21: UFilter BL.FinMeet B
    = UFilter BL.(FinMeet(B,id the carrier of BL)) by LATTICE4:def 9
      .= UFilter BL.((the L_meet of BL)$$(B,id the carrier of BL))
    by LATTICE2:def 4
      .= G$$(B,(UFilter BL)*(id the carrier of BL))
    by A17,A18,A19,A20,SETWISEO:30
      .= G$$(B,UFilter BL) by FUNCT_2:17
      .= G$$(B,(id bool ultraset BL)*UFilter BL) by FUNCT_2:17
      .= G$$(DD,id bool ultraset BL) by A17,A18,A19,SETWISEO:29;
    defpred X[Element of Fin bool ultraset BL] means
    for D being non empty Subset-Family of ultraset BL st
    D = $1 holds G$$($1,id bool ultraset BL) = meet D;
A22: X[{}.bool ultraset BL];
A23: for DD being (Element of Fin bool ultraset BL),
    b being Subset of ultraset BL st X[DD] holds X[DD \/ {.b.}]
    proof
      let DD be (Element of Fin bool ultraset BL), b be Subset of ultraset BL;
      assume
A24:  for D being non empty Subset-Family of ultraset BL st D = DD
      holds G$$(DD,id bool ultraset BL) = meet D;
      let D be non empty Subset-Family of ultraset BL such that
A25:  D = DD \/ {b};
      now per cases;
        suppose
A26:      DD is empty;
          hence G$$(DD \/ {.b.},id bool ultraset BL)=
          (id bool ultraset BL).b by A17,A18,SETWISEO:17
            .= b
            .= meet D by A25,A26,SETFAM_1:10;
        end;
        suppose
A27:      DD is non empty;
          DD c= D by A25,XBOOLE_1:7;
          then reconsider D1=DD as non empty Subset-Family of ultraset BL
          by A27,XBOOLE_1:1;
          reconsider D1 as non empty Subset-Family of ultraset BL;
          thus G$$(DD \/ {.b.},id bool ultraset BL) =
          G.(G$$(DD,id bool ultraset BL),(id bool ultraset BL).b)
          by A17,A18,A19,A27,SETWISEO:20
            .= G.(G$$(DD,id bool ultraset BL), b)
            .= G$$(DD,id bool ultraset BL) /\ b by A16
            .= meet D1 /\ b by A24
            .= meet D1 /\ meet {b} by SETFAM_1:10
            .= meet D by A25,SETFAM_1:9;
        end;
      end;
      hence thesis;
    end;
    for DD being Element of Fin bool ultraset BL holds X[DD]
    from SETWISEO:sch 4(A22,A23);
    then meet D = {}ultraset BL by A12,A15,A21;
    then
A28: union COMPLEMENT D = [#] ultraset BL \ {} by SETFAM_1:34
      .= ultraset BL;
A29: COMPLEMENT D c= X
    proof
      let x be object;
      assume
A30:  x in COMPLEMENT D;
      then reconsider c = x as Subset of ultraset BL;
      c` in D by A30,SETFAM_1:def 7;
      then consider a being object such that
A31:  a in dom UFilter BL and
A32:  a in B and
A33:  c` = UFilter BL.a by FUNCT_1:def 6;
      reconsider a as Element of BL by A31;
      a in CY by A11,A32;
      then a`` in CY;
      then consider b being Element of BL such that
A34:  b` = a`` and
A35:  UFilter BL.b in X;
      x = (UFilter BL.a)` by A33
        .= UFilter BL.a` by Th25
        .= UFilter BL.b`` by A34
        .= UFilter BL.b;
      hence thesis by A35;
    end;
    COMPLEMENT D is finite
    proof
A36:  D is finite;
      deffunc U(Subset of ultraset BL) = $1`;
A37:  { U(w) where w is Subset of ultraset BL : w in D } is finite
      from FRAENKEL:sch 21(A36);
      { w` where w is Subset of ultraset BL: w in D } = COMPLEMENT(D)
      proof
        hereby
          let x be object;
          assume x in { w` where w is Subset of ultraset BL : w in D };
          then consider w being Subset of ultraset BL such that
A38:      w` = x and
A39:      w in D;
          w`` in D by A39;
          hence x in COMPLEMENT(D) by A38,SETFAM_1:def 7;
        end;
        let x be object;
        assume
A40:    x in COMPLEMENT(D);
        then reconsider x9 = x as Subset of ultraset BL;
A41:    x9` in D by A40,SETFAM_1:def 7;
        x9``= x9;
        hence thesis by A41;
      end;
      hence thesis by A37;
    end;
    then COMPLEMENT D is Element of Fin X by A29,FINSUB_1:def 5;
    hence contradiction by A3,A28;
  end;
  then H <> the carrier of BL by Th28;
  then consider F such that
A42: H c= F and
A43: F is being_ultrafilter by FILTER_0:18;
A44: CY c= H by FILTER_0:def 4;
  not F in union X
  proof
    assume F in union X;
    then consider Y being set such that
A45: F in Y and
A46: Y in X by TARSKI:def 4;
    consider a being object such that
A47: a in dom UFilter BL and
A48: Y = UFilter BL.a by A2,A46,FUNCT_1:def 3;
    reconsider a as Element of BL by A47;
    a` in CY by A46,A48;
    then
A49: a` in H by A44;
    a in F by A45,A48,Th18;
    hence contradiction by A42,A43,A49,FILTER_0:46;
  end;
  hence contradiction by A1,A43;
end;
