Let's say, Bob has a death note. He gives up ownership, so that another one with shinigami eyes can tell him his lifespan. Bob is a genius, so he can calculate his death date. He writes his own name in it.
Bob Hyrule, Bradykardie, 2.3.2034.
He set his death date, by killing himself with a disease. Now he makes the shinigami eye deal. Will he die earlier?