Inverse Element in Inverse Completion of Commutative Monoid

Theorem
Let $\struct {S, \circ}$ be a commutative monoid.

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 the inverse of an element of $S$ which is invertible for $\circ$ is also its inverse for $\circ'$.

Proof
Let the identity of $\struct {S, \circ}$ be $e$.

Let $z$ be the inverse of $y$ for $\circ$:
 * $z \circ y = e$
 * $y \circ z = e$

From Identity of Inverse Completion of Commutative Monoid:
 * $z \circ' y = e$
 * $y \circ' z = e$

Hence $z$ is the inverse of $y$ for $\circ'$.