For all $1\leq i\leq m$,
\begin{align*}
\vectorentry{\vect{u}+\vect{v}}{i}
&=\vectorentry{\vect{u}}{i}+\vectorentry{\vect{v}}{i}
&&\knowl{./knowls/definition.CVA.knowl}{\text{Definition CVA}}\\
&=\vectorentry{\vect{v}}{i}+\vectorentry{\vect{u}}{i}
&&\text{Commutativity in $\complex{\null}$}\\
&=\vectorentry{\vect{v}+\vect{u}}{i}
&&\knowl{./knowls/definition.CVA.knowl}{\text{Definition CVA}}
\end{align*}
With equality of each component of the vectors $\vect{u}+\vect{v}$ and $\vect{v}+\vect{u}$ being equal Definition CVE tells us the two vectors are equal.