The way I see it, is ask yourself honestly if your neighbour asked you to pay for half their fence would you happily contribute, because I wouldn't.
We would because I am surprised that we don't seem to be responsible for any of the fences on our boundaries so it only seems fair to either share the cost or help maintain them. It's to our benefit too.
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and the fence is not only all over the shop with regard to the "horizontals" but is now leaning quite badly. 



