Subset Product is Subset of Generator

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

Let $$X, Y \subseteq \left({G, \circ}\right)$$.

Then $$X \circ Y \subseteq \left \langle {X, Y} \right \rangle$$ where:
 * $$X \circ Y$$ is the Subset Product of $$X$$ and $$Y$$ in $$G$$.
 * $$\left \langle {X, Y} \right \rangle$$ is the group generated by $X$ and $Y$.

Proof
It is clear from Set of Words Generates Group that $$W \left({\hat X \cup \hat Y}\right) = \left \langle {X, Y} \right \rangle$$.

It is equally clear that $$X \circ Y \subseteq W \left({\hat X \cup \hat Y}\right)$$.