Integer Addition is Associative

Theorem
The operation of addition on the set of integers $\Z$ is associative:
 * $\forall x, y, z \in \Z: x + \left({y + z}\right) = \left({x + y}\right) + z$

Proof
From the formal definition of integers, $\left[\!\left[{a, b}\right]\!\right]$ is an equivalence class of ordered pairs of natural numbers.

It can be taken directly from Additive Group of Integers‎ that the integers under addition form a group, from which associativity follows a priori.

Alternatively, it can be proved explicitly as follows.

Let $x = \left[\!\left[{a, b}\right]\!\right]$, $y = \left[\!\left[{c, d}\right]\!\right]$ and $z = \left[\!\left[{e, f}\right]\!\right]$ for some $x, y, z \in \Z$.

Then: