Extended Real Multiplication is Associative

Theorem
Extended real multiplication $\cdot_{\overline{\R}}$ is commutative.

That is, for all $x, y, z \in \overline{\R}$:


 * $(1):\quad x \cdot_{\overline{\R}} \left({y \cdot_{\overline{\R}} z}\right) = \left({x \cdot_{\overline{\R}} y}\right) \cdot_{\overline{\R}} z$

Proof
When $x,y,z \in \R$, then $(1)$ follows from Real Multiplication is Associative.

Next, the cases where at least one of $+\infty$ and $-\infty$ occurs need to be dealt with.