Before revision:  Herein, we prove that any quasi-left-continuous process satisfying NUPBR, will preserve the NUPBR when one stops it at any random horizon or when one incorporates a specific honest time.
After revision:  Herein, we prove that any quasi-left-continuous process satisfying NUPBR, will preserve the NUPBR when one stops it at any random horizon or when one incorporates a honest time.