1.13 $\forall a \in \mathbb{F}, a0=0$

Proof:
\begin{align*}
a0 &= a0 + 0 &&\text{(by 1.3)}\\
&= a0 + (a0 + (-a0)) &&\text{(by 1.4)}\\
&=(a0+a0) +(-a0)) &&\text{(by 1.2)}\\
&=a(0+0) +(-a0) &&\text{(by 1.9)}\\
&=a0+(-a0) &&\text{(by 1.3)}\\
&=0 &&\text{(by 1.4)}
\end{align*}