Integer Addition is Associative/Proof 2

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
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: