Subset Product is Subset of Generator

Theorem

Let $\struct {G, \circ}$ be a group.

Let $X, Y \subseteq \struct {G, \circ}$.

Then $X \circ Y \subseteq \gen {X, Y}$ where:

$X \circ Y$ is the Subset Product of $X$ and $Y$ in $G$.
$\gen {X, Y}$ is the subgroup of $G$ generated by $X$ and $Y$.

Proof

It is clear from Set of Words Generates Group that $\map W {\hat X \cup \hat Y} = \gen {X, Y}$.

It is equally clear that $X \circ Y \subseteq \map W {\hat X \cup \hat Y}$.

$\blacksquare$