reserve X for non empty TopSpace,
  D for Subset of X;

theorem Th3:
  for C being Subset of X modified_with_respect_to D` st C = D
  holds C is closed
proof
  let C be Subset of X modified_with_respect_to D`;
  assume C = D;
  then C` = D` by TMAP_1:93;
  then C` is open by TMAP_1:94;
  hence thesis;
end;
