Product Inverse Operation Properties/Lemma 1

Theorem
Let $\struct {G, \oplus}$ be a closed algebraic structure on which the following properties hold:

Let $\circ$ be the operation on $G$ defined as:
 * $\forall x, y \in G: x \circ y = x \oplus \paren {e \oplus y}$

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