Quotient Structure is Well-Defined

Theorem
Let $\left({S, \circ}\right)$ be an algebraic structure.

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

Let $S / \mathcal R$ be the quotient set of $S$ by $\mathcal R$.

Let $\circ_\mathcal R$ be the operation induced on $S / \mathcal R$ by $\circ$.

Then $\circ_\mathcal R$ is a well-defined operation in the quotient structure $\left({S / \mathcal R, \circ_\mathcal R}\right)$.