Quotient Structure of Monoid is Monoid

Theorem
Let $\RR$ be a congruence relation on a monoid $\struct {S, \circ}$ with an identity $e$.

Then the quotient structure $\struct {S / \RR, \circ_\RR}$ is a monoid.

Proof
From Quotient Structure of Semigroup is Semigroup $\struct {S / \RR, \circ_\RR}$ is a semigroup.

Let $\eqclass x {\RR} \in S / \RR$.

Consider $\eqclass e \RR$:

Furthermore:

Hence $\eqclass e \RR$ is an identity.

Hence $\struct {S / \RR, \circ_\RR}$ is a monoid.