Integer Multiplication is Associative

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


 * $\forall x, y, z \in \Z: x \times \paren {y \times z} = \paren {x \times y} \times z$

Proof
From the formal definition of integers, $\eqclass {a, b} {}$ is an equivalence class of ordered pairs of natural numbers.

Let $x = \eqclass {a, b} {}$, $y = \eqclass {c, d} {}$ and $z = \eqclass {e, f} {}$ for some $x, y, z \in \Z$.

Then: