Product of Subgroup with Inverse

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

Then:
 * $\forall H \le \struct {G, \circ}:$
 * $H^{-1} \circ H = H$
 * $H \circ H^{-1} = H$

where $H \le G$ denotes that $H$ is a subgroup of $G$.

Proof
From Inverse of Subgroup:
 * $H = H^{-1}$

From Product of Subgroup with Itself:
 * $H \circ H = H$

The result follows.