I picked up Spivak's Calculus (3rd Edition) today and it seemed like a good idea to go through the section Basic Properties of Numbers. In this chapter, Spivak proves that
$a \cdot 0 = 0$
The proof looks simple:
$a \cdot 0 + a \centerdot 0 = a \centerdot (0+0) = a \centerdot 0$
My question is just as simple: How do I get from $a \centerdot 0 = 0$ to $a \centerdot 0 + a \centerdot 0 = a \centerdot (0+0)$?