Lemma Suppose $(G,\ast)$ is a group. Then every element in $G$ has a unique inverse.

Proof. Suppose $g\in G$ . By the group axioms we know that there is an $h\in G$ such that $$ g\ast h = h\ast g = e,$$ where $e$ is the identity element in $G$ . If there is also a $h'\in G$ satisfying $$ g\ast h' = h'\ast g = e,$$ then $$ h = h\ast e = h \ast(g \ast h') =(h\ast g)\ast h' = e\ast h' = h',$$ so $h=h'$ , and $g$ has a unique inverse.