Construction of Inverse Completion/Quotient Structure is Commutative Semigroup

Theorem
Then:
 * $\left({T', \oplus'}\right)$ is a commutative semigroup.

Proof
The canonical epimorphism from $\left({S \times C, \oplus}\right)$ onto $\left({T\,', \oplus\,'}\right)$ is given by:


 * $q_\boxtimes: \left({S \times C, \oplus}\right) \to \left({T\,', \oplus\,'}\right): q_\boxtimes \left({x, y}\right) = \left[\!\left[{\left({x, y}\right)}\right]\!\right]_\boxtimes$

where, by definition:

By Morphism Property Preserves Closure, as $\oplus$ is closed, then so is $\oplus\,'$.

By Epimorphism Preserves Associativity, as $\oplus$ is associative, then so is $\oplus\,'$.

By Epimorphism Preserves Commutativity, as $\oplus$ is commutative, then so is $\oplus\,'$.

Thus $\left({T\,', \oplus\,'}\right)$ is closed, associative and commutative, and therefore a commutative semigroup.