Integer Multiplication is Associative

Theorem
The operation of multiplication on the set of integers $\Z$ is associative:


 * $\forall x, y, z \in \Z: x \times \left({y \times z}\right) = \left({x \times y}\right) \times z$

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

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: