Take a vector v \in V and additive inverses a,b \in V.
\begin{equation} a+0 = a \end{equation}
defn. of additive identity
\begin{equation} a+(v+b) = a \end{equation}
defn. of additive inverse
\begin{equation} (a+v)+b = a \end{equation}
associativity
\begin{equation} 0+b = a \end{equation}
defn. of additive inverse
\begin{equation} b=a\ \blacksquare \end{equation}