Product of Subset with Intersection/Proof 2

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

Let $X, Y, Z \subseteq G$.

Then:

where $X \circ Y$ denotes the subset product of $X$ and $Y$.
 * $X \circ \left({Y \cap Z}\right) \subseteq \left({X \circ Y}\right) \cap \left({X \circ Z}\right)$
 * $\left({Y \cap Z}\right) \circ X \subseteq \left({Y \circ X}\right) \cap \left({Z \circ X}\right)$

Proof
Consider the relation $\mathcal R \subseteq G \times G$ defined as:


 * $\forall g, h \in G: \left({g, h}\right) \in \mathcal R \iff \exists g \in X$

Then:
 * $\forall S \subseteq G: X \circ S = \mathcal R \left({S}\right)$

Then:

Next, consider the relation $\mathcal R \subseteq G \times G$ defined as:


 * $\forall g, h \in G: \left({g, h}\right) \in \mathcal R \iff \exists h \in X$

Then:
 * $\forall S \subseteq G: S \circ X = \mathcal R \left({S}\right)$

Then: