Inverse Completion is Commutative Semigroup

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

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 $T = S \circ' C^{-1}$, and is a commutative semigroup.

Proof
From Subset Product defining Inverse Completion of Commutative Semigroup is Commutative Semigroup:
 * $S \circ' C^{-1}$ is a commutative semigroup.

From Structure of Inverse Completion of Commutative Semigroup:
 * $T = S \circ' C^{-1}$