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

theorem Th9:
  for Q being Subset of T holds Q in Kurat7Set A implies Int Q in
  Kurat7Set A & Cl Q in Kurat7Set A
proof
  let Q be Subset of T;
  assume
A1: Q in Kurat7Set A;
  Int Q in Kurat7Set A & Cl Q in Kurat7Set A
  proof
    per cases by A1,ENUMSET1:def 5;
    suppose
      Q = A;
      hence thesis by ENUMSET1:def 5;
    end;
    suppose
      Q = Int A;
      hence thesis by ENUMSET1:def 5;
    end;
    suppose
      Q = Cl A;
      hence thesis by ENUMSET1:def 5;
    end;
    suppose
      Q = Int Cl A;
      hence thesis by ENUMSET1:def 5;
    end;
    suppose
      Q = Cl Int A;
      hence thesis by ENUMSET1:def 5;
    end;
    suppose
A2:   Q = Cl Int Cl A;
      Int Cl Int Cl A = Int Cl A by TDLAT_1:5;
      hence thesis by A2,ENUMSET1:def 5;
    end;
    suppose
A3:   Q = Int Cl Int A;
      then Cl Q = Cl Int A by TOPS_1:26;
      hence thesis by A3,ENUMSET1:def 5;
    end;
  end;
  hence thesis;
end;
