Product Inverse Operation Properties/Lemma 4

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:
 * $\forall x, y, z \in G: x \oplus z = y \oplus z \implies x = y$

Proof
Let $x \oplus z = y \oplus z$.

Then we have:

Then:

Thus we have:
 * $x \oplus y = e$

as both are equal to $\paren {x \oplus z} \oplus \paren {y \oplus z}$.

Then from Lemma $3$:
 * $x = y$

Hence the result.