Identity of Inverse Completion of Commutative Monoid

Theorem
Let $\struct {S, \circ}$ be a commutative monoid whose identity is $e$.

Let $\struct {C, \circ} \subseteq \struct {S, \circ}$ be the subsemigroup of cancellable elements of $\struct {S, \circ}$.

Let $\struct {T, \circ'}$ be an inverse completion of $\struct {S, \circ}$.

Then $e \in T$ is the identity for $\circ'$.

Proof
Let $e$ be the identity for $\circ$.

Let $e = x \circ' y^{-1}$, where $x \in S, y \in C$.

Then:

Thus $e = y^{-1} \circ' y$, and $y^{-1} \circ' y$ is the identity for $\circ'$.