Proof of Theorem ZPZT

(⇒)We conduct the forward argument in two cases. First suppose that $\alpha=0$. Then we are done. (That was easy.)

For the second case, suppose now that $\alpha\neq 0$. Then \begin{align*} \beta&=1\beta&&\knowl{./knowls/property.OCN.knowl}{\text{Property OCN}}\\ &=\left(\frac{1}{\alpha}\alpha\right)\beta&&\knowl{./knowls/property.MICN.knowl}{\text{Property MICN}}\\ &=\frac{1}{\alpha}\left(\alpha\beta\right)&&\knowl{./knowls/property.MACN.knowl}{\text{Property MACN}}\\ &=\frac{1}{\alpha}0&&\text{Hypothesis}\\ &=0&&\knowl{./knowls/theorem.ZPCN.knowl}{\text{Theorem ZPCN}} \end{align*}

(⇐)With two applications of Theorem ZPCN it is easy to see that if one of the scalars is zero, then so is the product.