The inverse in a Group is unique
# Statement
Suppose $G$ is a Group and $u, v, w \in G$ so that $v$, $w$ are both Additive Inverses of $u$. Then $v = w$.
# Proof
Denote $e \in G$ the Additive Identity. Then $$w = we = w(uv) = (wu)v = ev = v$$ by Associativity. $\blacksquare$