reserve T for non empty TopSpace,
  X,Z for Subset of T;
reserve x,y for Element of OpenClosedSet(T);

theorem Th8:
  x` is Element of OpenClosedSet(T)
proof
  reconsider y = x as Subset of T;
A1: y is open by Th1;
  y is closed by Th2;
  then
A2: x` is open;
  x` is closed by A1;
  hence thesis by A2,Th3;
end;
