Group Identity is Right Identity for Product Inverse Operation

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

Let $\oplus: G \times G \to G$ be the product inverse of $\circ$ on $G$.

Then $e$ is also the right identity of $\struct {G, \oplus}$, in the sense that:
 * $\forall x \in G: x \oplus e = x$