Pointwise Inverse in Induced Structure

Theorem
Let $$\left({T, \oplus}\right)$$ be a group whose identity is $$e_T$$, and let $$S$$ be a set.

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

Let $$f \in T^S$$.

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


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

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

Proof
Let $$f \in T^S$$.

$$ $$ $$

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