Quotient Structure on Subset Product

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

Let $\mathcal R$ be a congruence for $\circ$ on $S$.

Then:
 * $\forall X, Y \in S / \mathcal R: X \circ_\mathcal P Y \subseteq X \circ_\mathcal R Y$

where:


 * $S / \mathcal R$ is the quotient of $S$ by $\mathcal R$


 * $\circ_\mathcal P$ is the operation induced on $\mathcal P \left({S}\right)$ by $\circ$


 * $\circ_\mathcal R$ is the operation induced on $S / \mathcal R$ by $\circ$

Proof
By definition of subset product:


 * $X \circ_\mathcal P Y = \left\{{x \circ y: x \in X, y \in Y}\right\}$

Thus:


 * $X \circ_\mathcal R Y = \left\{{x \circ y: x \in X, y \in Y}\right\} \cup \left\{{x \circ y: x \in \left[\!\left[{X}\right]\!\right]_\mathcal R, y \in \left[\!\left[{Y}\right]\!\right]_\mathcal R}\right\}$

The result follows from Subset of Union.