Quotient Structure of Semigroup is Semigroup

Theorem
Let $\RR$ be a congruence relation on a semigroup $\struct {S, \circ}$.

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

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

Let $\eqclass x \RR, \eqclass y \RR, \eqclass z \RR \in S / \RR$.

We shall prove that $\circ_\RR$ is associative:

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