Show in a group $G$ that $(a^{-1})^{-1} = a $.
The proof looks fine. However, since you make a point of both mentioning and proving that each element has a unique inverse I would advise you to make precise what you mean when you write $a^{-1}$.
The proof looks fine. However, since you make a point of both mentioning and proving that each element has a unique inverse I would advise you to make precise what you mean when you write $a^{-1}$.