Extended Real Multiplication is Associative
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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.
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