Product of Subset with Intersection

Theorem
Let $\struct {G, \circ}$ be an algebraic structure.

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

Then:


 * $X \circ \paren {Y \cap Z} \subseteq \paren {X \circ Y} \cap \paren {X \circ Z}$


 * $\paren {Y \cap Z} \circ X \subseteq \paren {Y \circ X} \cap \paren {Z \circ X}$

where $X \circ Y$ denotes the subset product of $X$ and $Y$.