Integer Multiplication is Commutative

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


 * $\forall x, y \in \Z: x \times y = y \times x$

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

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

Then: