Extended Real Multiplication is Commutative

Theorem

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

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

$x \times_{\overline \R} y = y \times_{\overline \R} x$

Proof

Let $x, y \in \R$.

Then from Real Multiplication is Commutative:

$x \times_{\overline \R} y = y \times_{\overline \R} x$

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

$\blacksquare$