I am going through Analysis V.1, Terence Tao. In his definition of Addition, screenshot given below, how did he deduce that (N++)+M := (N+M)++ ?? I am not able to understand the steps.
You have to remember that in developing formal rules for addition, we are taking the properties in which we are already naturally familiar and changing them into a rigorous definition.
Informally, we perceive $n++$ as $n+1$. Therefore the recursive definition simply says that $$(n+1) + m = (n+m)+1$$ You must remember that we already have in mind what properties we want addition to have and that we are simply providing a definition.