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*}