Pointwise Inverse in Induced Structure

Theorem
Let $\struct {G, \oplus}$ be a group whose identity is $e_G$.

Let $S$ be a set.

Let $\struct {G^S, \oplus}$ be the structure on $G^S$ induced by $\oplus$.

Let $f \in G^S$.

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


 * $\forall f \in G^S: \forall x \in S: \map {f^*} x = \paren {\map f x}^{-1}$

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

Proof
Let $f \in G^S$.

Similarly for $\map {\paren {f^* \oplus f} } x$.