Prove that $0{\bf v}={\bf 0}$ for any vector ${\bf v}\in V$.
Solution: We have
\begin{align*}
0{\bf v}=&\ 0{\bf v}+{\bf 0}\\ =&\ 0{\bf v}+({\bf v}+(-{\bf v}))\\
=&\ (0{\bf v}+{\bf v})+(-{\bf v})\\ =&\ (0{\bf v}+1{\bf v})+(-{\bf v})\\
=&\ (0+1){\bf v}+(-{\bf v})\\=&\ 1{\bf v}+(-{\bf v})\\ =&\ {\bf v}+(-{\bf v})={\bf 0}.
\end{align*}