1.16 If $ab=0$, then either $a=0$ or $b=0$ (or both)

Proof:

Assume $a \neq 0$. Then
\begin{align*}
b &= b1 && \text{(by 1.7)} \\
&=b(ac) && \text{(by 1.8)} \\
&=(ba)c && \text{(by 1.6)} \\
&=(ab)c && \text{(by 1.5)} \\
&=0c && \text{(by the assumption)} \\
&=c0 && \text{(by 1.5)} \\
&=0 && \text{(by 1.13)}
\end{align*}