reserve T for non empty TopSpace;
reserve A for Subset of T;

theorem Th6:
  for Q being Subset of T holds Q in Kurat14Set A implies Q` in
  Kurat14Set A & Q- in Kurat14Set A
proof
  let Q be Subset of T;
  set k1 = Cl A, k2 = (Cl A)`, k3 = Cl (Cl A)`, k4 = (Cl (Cl A)`)`, k5 = Cl (
Cl (Cl A)`)`, k6 = (Cl (Cl (Cl A)`)`)`, k7 = Cl A`, k8 = (Cl A`)`, k9 = Cl (Cl
A`)`, k10 = (Cl (Cl A`)`)`, k11 = Cl (Cl (Cl A`)`)`, k12 = (Cl (Cl (Cl A`)`)`)`
  ;
  assume
A1: Q in Kurat14Set A;
  per cases by A1,XBOOLE_0:def 3;
  suppose
A2: Q in { A, k1, k2, k3, k4, k5, k6 };
    Q` in Kurat14Set A & Cl Q in Kurat14Set A
    proof
      per cases by A2,ENUMSET1:def 5;
      suppose
        Q = A;
        then
        Cl Q in { A, Cl A, (Cl A)`, Cl (Cl A)`, (Cl (Cl A)`)`, Cl (Cl (Cl
A)`)`, (Cl (Cl (Cl A)`)`)` } & Q` in { A`, Cl A`, (Cl A`)`, Cl (Cl A`)`, (Cl (
        Cl A`)`)`, Cl (Cl (Cl A`)`)`, (Cl (Cl (Cl A`)`)`)` } by ENUMSET1:def 5;
        hence thesis by XBOOLE_0:def 3;
      end;
      suppose
        Q = Cl A;
        then
        Cl Q in { A, Cl A, (Cl A)`, Cl (Cl A)`, (Cl (Cl A)`)`, Cl (Cl (Cl
        A)`)`, (Cl (Cl (Cl A)`)`)` } & Q` in { A, k1, k2, k3, k4, k5, k6 } by
ENUMSET1:def 5;
        hence thesis by XBOOLE_0:def 3;
      end;
      suppose
        Q = (Cl A)`;
        then
        Cl Q in { A, Cl A, (Cl A)`, Cl (Cl A)`, (Cl (Cl A)`)`, Cl (Cl (Cl
A)`)`, (Cl (Cl (Cl A)`)`)` } & Q` in { A, Cl A, (Cl A)`, k3, k4, k5, k6 } by
ENUMSET1:def 5;
        hence thesis by XBOOLE_0:def 3;
      end;
      suppose
        Q = Cl (Cl A)`;
        then
        Cl Q in { A, Cl A, (Cl A)`, Cl (Cl A)`, (Cl (Cl A)`)`, Cl (Cl (Cl
        A)`)`, (Cl (Cl (Cl A)`)`)` } & Q` in { A, k1, k2, k3, k4, k5, k6 } by
ENUMSET1:def 5;
        hence thesis by XBOOLE_0:def 3;
      end;
      suppose
        Q = (Cl (Cl A)`)`;
        then
        Cl Q in { A, Cl A, (Cl A)`, Cl (Cl A)`, (Cl (Cl A)`)`, Cl (Cl (Cl
A)`)`, (Cl (Cl (Cl A)`)`)` } & Q` in { A, k1, k2, Cl (Cl A)`, k4, k5, k6 } by
ENUMSET1:def 5;
        hence thesis by XBOOLE_0:def 3;
      end;
      suppose
        Q = Cl (Cl (Cl A)`)`;
        then
        Cl Q in { A, Cl A, (Cl A)`, Cl (Cl A)`, (Cl (Cl A)`)`, Cl (Cl (Cl
        A)`)`, (Cl (Cl (Cl A)`)`)` } & Q` in { A, k1, k2, k3, k4, k5, k6 } by
ENUMSET1:def 5;
        hence thesis by XBOOLE_0:def 3;
      end;
      suppose
A3:     Q = (Cl (Cl (Cl A)`)`)`;
        Cl (Cl (Cl (Cl A)`)`)` = Cl (Cl A)` by Th1;
        then
A4:     Cl Q in { A, Cl A, (Cl A)`, Cl (Cl A)`, (Cl (Cl A)`)`, Cl (Cl (Cl
        A)`)`, (Cl (Cl (Cl A)`)`)` } by A3,ENUMSET1:def 5;
        Q` in { A, k1, k2, k3, k4, k5, k6 } by A3,ENUMSET1:def 5;
        hence thesis by A4,XBOOLE_0:def 3;
      end;
    end;
    hence thesis;
  end;
  suppose
A5: Q in { A`, k7, k8, k9, k10, k11, k12 };
    Q` in Kurat14Set A & Cl Q in Kurat14Set A
    proof
      per cases by A5,ENUMSET1:def 5;
      suppose
        Q = A`;
        then
        Cl Q in { A`, Cl A`, k8, k9, k10, k11, k12 } & Q` in { A, k1, k2,
        k3, k4, k5, k6 } by ENUMSET1:def 5;
        hence thesis by XBOOLE_0:def 3;
      end;
      suppose
        Q = Cl A`;
        then
        Cl Q in { A`, Cl A`, k8, k9, k10, k11, k12} & Q` in { A`, k7, (Cl
        A`)`, k9, k10, k11, k12} by ENUMSET1:def 5;
        hence thesis by XBOOLE_0:def 3;
      end;
      suppose
        Q = (Cl A`)`;
        then Cl Q in { A`, k7, k8, Cl (Cl A`)`, k10, k11, k12} & Q` in { A`,
        Cl A`, k8, k9, k10, k11, k12} by ENUMSET1:def 5;
        hence thesis by XBOOLE_0:def 3;
      end;
      suppose
        Q = Cl (Cl A`)`;
        then
        Cl Q in { A`, k7, k8, k9, k10, k11, k12} & Q` in { A`, k7, k8, k9
        , k10, k11, k12} by ENUMSET1:def 5;
        hence thesis by XBOOLE_0:def 3;
      end;
      suppose
        Q = (Cl (Cl A`)`)`;
        then
        Cl Q in { A`, k7, k8, k9, k10, k11, k12} & Q` in { A`, k7, k8, k9
        , k10, k11, k12} by ENUMSET1:def 5;
        hence thesis by XBOOLE_0:def 3;
      end;
      suppose
        Q = Cl (Cl (Cl A`)`)`;
        then
        Cl Q in { A`, k7, k8, k9, k10, k11, k12} & Q` in { A`, k7, k8, k9
        , k10, k11, k12} by ENUMSET1:def 5;
        hence thesis by XBOOLE_0:def 3;
      end;
      suppose
A6:     Q = (Cl (Cl (Cl A`)`)`)`;
        then Cl Q = Cl (Cl A`)` by Th1;
        then
A7:     Cl Q in { A`, k7, k8, k9, k10, k11, k12} by ENUMSET1:def 5;
        Q` in { A`, k7, k8, k9, k10, k11, k12} by A6,ENUMSET1:def 5;
        hence thesis by A7,XBOOLE_0:def 3;
      end;
    end;
    hence thesis;
  end;
end;
