What would you do?

You rent a house with a wooden fence (belonging to the rental property, not the neighbors).   It's falling apart, it's so rotten the boards can't be nailed back up, and you have to prop up other random pieces of boards and such to cover the gaps, as do the neighbors.  You have a dog and the neighbors on the other side of the fence has a dog.  The neighbor's dog randomly sticks it's head through the fence and barks at your dog.  At some point, your dog gets bitten.  Nobody saw what actually happened, or whether or not the bite was provoked.   You take your (pure bred) dog to the vet and have to pay for stitches and random shots because you don't know the health of the neighbors dog. 
 
OK, now let's turn it around..
You are the neighbor.  It was your dog that injured the other dog.  But since nobody saw what actually happened, you don't know if it was provoked or even if your dog actually bit the other dog.  For all you know, the neighbor's dog got it's head stuck in the broken fence.  But you do know for a fact that your dog does have a tendancy to stick it's head through that part of the fence.  The neighbor has let you know what has happened.  Then later comes back with a bill explaining that they are paying for the random shots (which aren't cheap, but were voluntary), but would like you to pay for the stitches and the emergency visit. 
 
In either case, what would you do?       

ETA since everybody's focused on the fence.
The broken ass fence will be replaced in a couple of months when our tax rebate comes.  It's been a priority for a while, but it's gotten much worse over the winter.  We can't afford it until then.