Product of Subset with Intersection/Proof 1

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

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

Consider $X \circ \paren {Y \cap Z}$.

We have $x \circ t \in X \circ \paren {Y \cap Z}$ 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 $\paren {Y \cap Z} \circ X$.

Then we have $t \circ x \in \paren {Y \cap Z} \circ X$ 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.