Extended Real Multiplication is Commutative

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

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


 * $x \cdot_{\overline{\R}} y = y \cdot_{\overline{\R}} x$

Proof
When $x,y \in \R$, then $x \cdot_{\overline{\R}} y = y \cdot_{\overline{\R}} x$ follows from Real Multiplication is Commutative.

The remaining cases are explicitly imposed in the definition of $\cdot_{\overline{\R}}$.