Product of Subgroup with Itself

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

Then:
 * $\forall H \le G: H \circ H = H$

where $\le$ denotes the subgroup relation.

Proof
From Magma Subset Product with Self, we have:
 * $H \circ H \subseteq H$

Let $e$ be the identity of $G$.

By Identity of Subgroup, it is also the identity of $H$.

So:

Hence the result from the definition of set equality.