Quotient Structure of Semigroup is Semigroup

Theorem
Let $\mathcal R$ be a congruence relation on a semigroup $\left({S, \circ}\right)$.

Then the quotient structure $\left({S / \mathcal R, \circ_\mathcal R}\right)$ is a semigroup.

Proof
From Quotient Structure is Well-Defined we have that $\circ_\mathcal R$ is closed on $S / \mathcal R$.

Let $\left[\!\left[{x}\right]\!\right]_\mathcal R, \left[\!\left[{y}\right]\!\right]_\mathcal R, \left[\!\left[{z}\right]\!\right]_\mathcal R \in S / \mathcal R$.

We shall prove that $\circ_\mathcal R$ is associative:

Hence $\left({S / \mathcal R, \circ_\mathcal R}\right)$ is a semigroup.