Integer Multiplication is Associative

Theorem
Integer multiplication 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:

$$ $$ $$ $$ $$ $$ $$ $$ $$