Equivalence Relation inducing Closed Quotient Set of Magma is Congruence Relation

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

Let $\circ_\PP$ be the operation induced on $\powerset S$, the power set of $S$.

Let $\RR$ be an equivalence relation on $S$.

Let $S / \RR$ denote the quotient set of $S$ induced by $\RR$.

Let the algebraic structure $\struct {S / \RR, \circ_\PP}$ be closed.

Then:
 * $\RR$ is a congruence relation for $\circ$

and:
 * the operation $\circ_\RR$ induced on $S / \RR$ by $\circ$ is the operation induced on $S / \RR$ by $\circ_\PP$.

Proof
Let $x_1, y_1, x_2, y_2 \in S$ be arbitrary, such that:
 * $x_1 \mathrel \RR x_2$
 * $y_1 \mathrel \RR y_2$

To demonstrate that $\RR$ is a congruence relation for $\circ$, we need to show that:
 * $\paren {x_1 \circ y_1} \mathrel \RR \paren {x_2 \circ y_2}$