I've gone through the similar post [Modulo of a negative number](https://math.stackexchange.com/q/519845/103816) . But that post is not about proof and I'm asking for the proof in general.

This question is another [follow up question](https://math.stackexchange.com/questions/1377221/why-do-remainders-show-cyclic-pattern#comment2805043_1377241) of my [previous question](https://math.stackexchange.com/q/1377221/103816). The question is how can we prove that $(a-b) \mod N = a \mod N + ((-b) \mod N)$.