Pointwise Inverse in Induced Structure

Theorem
Let $\left({G, \oplus}\right)$ be a group whose identity is $e_G$.

Let $S$ be a set.

Let $\left({G^S, \oplus}\right)$ be the structure on $G^S$ induced by $\oplus$.

Let $f \in G^S$.

Let $f^* \in G^S$ be defined as follows:


 * $\forall f \in G^S: \forall x \in S: f^* \left({x}\right) = \left({f \left({x}\right)}\right)^{-1}$

Then $f^*$ is the inverse of $f$ for the operation induced on $G^S$ by $\oplus$.

Proof
Let $f \in G^S$.

... and similarly for $\left({f^* \oplus f}\right) \left({x}\right)$.