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.