Cancellation Property of Product Inverse Operator

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:
 * $\forall x, y, z \in G: \paren {x \oplus z} \oplus \paren {y \oplus z} = x \oplus y$