reserve x,y,X for set;

theorem Th31:
  for T being non empty TopSpace holds T is compact iff for F
  being ultra Filter of BoolePoset [#]T ex x being Point of T st x
  is_a_convergence_point_of F, T
proof
  let T be non empty TopSpace;
  set X = the carrier of T;
  hereby
    assume
A1: T is compact;
    let F be ultra Filter of BoolePoset [#]T;
    set G = {Cl A where A is Subset of T: A in F};
    G c= bool the carrier of T
    proof
      let x be object;
      assume x in G;
      then ex A being Subset of T st x = Cl A & A in F;
      hence thesis;
    end;
    then reconsider G as Subset-Family of T;
A2: G is centered
    proof
      set A0 = the Element of F;
      reconsider A0 as Subset of T by WAYBEL_7:2;
      Cl A0 in G;
      hence G <> {};
      let H be set;
      assume that
A3:   H <> {} and
A4:   H c= G and
A5:   H is finite;
      reconsider H1 = H as finite Subset-Family of X by A4,A5,XBOOLE_1:1;
      H1 c= F
      proof
        let x be object;
        assume x in H1;
        then x in G by A4;
        then consider A being Subset of T such that
A6:     x = Cl A and
A7:     A in F;
        A c= Cl A by PRE_TOPC:18;
        hence thesis by A6,A7,WAYBEL_7:7;
      end;
      then Intersect H1 in F by WAYBEL_7:11;
      then Intersect H1 <> {} by Th1;
      hence thesis by A3,SETFAM_1:def 9;
    end;
    set x = the Element of meet G;
    G is closed
    proof
      let A be Subset of T;
      assume A in G;
      then ex B being Subset of T st A = Cl B & B in F;
      hence thesis;
    end;
    then
A8: meet G <> {} by A1,A2,COMPTS_1:4;
    then x in meet G;
    then reconsider x as Point of T;
    take x;
    thus x is_a_convergence_point_of F, T
    proof
      let A be Subset of T such that
A9:   A is open and
A10:  x in A;
      set B = A`;
A11:  now
        assume B in F;
        then Cl B in G;
        then
A12:    B in G by A9,PRE_TOPC:22;
        not x in B by A10,XBOOLE_0:def 5;
        hence contradiction by A8,A12,SETFAM_1:def 1;
      end;
      F is prime by WAYBEL_7:22;
      hence thesis by A11,WAYBEL_7:21;
    end;
  end;
  assume
A13: for F being ultra Filter of BoolePoset [#]T ex x being Point of T
  st x is_a_convergence_point_of F, T;
  now
    set L = BoolePoset X;
    let F be Subset-Family of T such that
A14: F is centered closed;
    reconsider Y = F as Subset of BoolePoset X by WAYBEL_7:2;
    set G = uparrow fininfs Y;
    now
      assume Bottom L in G;
      then consider x being Element of BoolePoset X such that
A15:  x <= Bottom L and
A16:  x in fininfs Y by WAYBEL_0:def 16;
A17:  Bottom L = {} by YELLOW_1:18;
      consider Z being finite Subset of Y such that
A18:  x = "/\"(Z,L) and
      ex_inf_of Z,L by A16;
      reconsider Z as Subset of L by XBOOLE_1:1;
A19:  x = Bottom L by A15,YELLOW_5:19;
      then x <> Top L by WAYBEL_7:3;
      then
A20:  Z <> {} by A18,YELLOW_0:def 12;
      then meet Z <> {} by A14;
      hence contradiction by A17,A18,A19,A20,YELLOW_1:20;
    end;
    then G is proper;
    then consider UF being Filter of L such that
A21: G c= UF and
A22: UF is ultra by WAYBEL_7:26;
    consider x being Point of T such that
A23: x is_a_convergence_point_of UF, T by A13,A22;
    F c= G by WAYBEL_0:62;
    then
A24: F c= UF by A21;
A25: now
      let A be set;
      assume
A26:  A in F;
      then reconsider B = A as Subset of T;
A27:  now
        let C be Subset of T;
        assume that
A28:    C is open and
A29:    x in C;
A30:    C in UF by A23,A28,A29;
        A in UF by A24,A26;
        then reconsider c = C, a = A as Element of L by A30;
        a"/\"c in UF by A24,A26,A30,WAYBEL_0:41;
        then a"/\"c <> {} by A22,Th1;
        then A /\ C <> {} by YELLOW_1:17;
        hence A meets C;
      end;
      B is closed by A14,A26;
      then Cl B = B by PRE_TOPC:22;
      hence x in A by A27,PRE_TOPC:24;
    end;
    F <> {} by A14;
    hence meet F <> {} by A25,SETFAM_1:def 1;
  end;
  hence thesis by COMPTS_1:4;
end;
