How can a death note shorten the lifespan of others? The use of the Death Note in the human world sometimes affects other humans' lives or shortens their original life span, even though their names are not actually written in the Death Note itself. In these cases, no matter the cause, the god of death sees only the original lifespan and not the shortened lifespan.

We saw, that Misas lifespan was shortened because Light was killed by Ryuk. So this means, that you can shorten a persons lifespan with the death note. So let's say, A has a death note and knows that C wants to kill B with a death note. He writes B's name in his death note, thus preventing him from being killed by a death note. So C, can't kill B with a death note anymore. But since the death note can shorten the lifespan of those, whos name isn't written in the death note, C decides to use Y, to set up a trap. C writes:

Y, Car accident. Dies at 12:30, after he produces a deadly poison, which he injects into the food the person he knows as ''B'' eats. After he did this, he will die 9 days later in a car accident.

So, could C shorten B's lifespan in this way, so that he is able to indirectly kill him?
Yes, to the extent of how can you manipulate a person's actions with the DN. (1) (2). But care must be taken not to break the other rules:

In your example, B will die in at most 23 days because A has written B's name in his DN.

But let's assume C wants B dead BEFORE that timeframe. 

There is a conflict in the writing C did in the DN:


  Y, Car accident. Dies at 12:30, after he produces a deadly poison, which he injects into the food the person he knows as ''B'' eats. After he did this, he will die 9 days later in a car accident.


C's writing may be in conflict with this rule:


  If the death leads to the death of more than the intended, the person will simply die of a heart attack. This is to ensure that other lives are not influenced. Even though only one name is written in the Death Note, if it influences and causes other humans that are not written in it to die, the victim’s cause of death will be a heart attack


So it can either be considered as a violation of the rule above (and Y will die of heart attack) or work as intended by C.