Product of Subset with Intersection

Theorem
Let $$\left({G, \circ}\right)$$ be a group.

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

Then:


 * $$X \circ \left({Y \cap Z}\right) \subseteq \left({X \circ Y}\right) \cap \left({X \circ Z}\right)$$
 * $$\left({Y \cap Z}\right) \circ X \subseteq \left({Y \circ X}\right) \cap \left({Z \circ X}\right)$$

Proof
Let $$x \in X, t \in Y \cap Z$$.

By the definition of intersection, $$t \in Y$$ and $$t \in Z$$.


 * Consider $$X \circ \left({Y \cap Z}\right)$$.

We have $$x \circ t \in X \circ \left({Y \cap Z}\right)$$ by definition of subset product.

As $$t \in Y$$ and $$t \in Z$$, we also have $$x \circ t \in X \circ Y$$ and $$x \circ t \in X \circ Z$$.

The result follows.


 * Similarly, consider $$\left({Y \cap Z}\right) \circ X$$.

Then we have $$t \circ x \in \left({Y \cap Z}\right) \circ X$$ by definition of subset product.

As $$t \in Y$$ and $$t \in Z$$, we also have $$t \circ x \in Y \circ X$$ and $$t \circ x \in Z \circ X$$.

Again, the result follows.