Inverses in Subgroup

Theorem
Let $G$ be a group.

Let $H$ be a subgroup of $G$.

Then for each $h \in H$, the inverse of $h$ in $H$ is the same as the inverse of $h$ in $G$.

Proof
Let $h \in H$.

Let:
 * $h'$ be the inverse of $h$ in $H$:
 * $h^{-1}$ be the inverse of $h$ in $G$.

From Identity of Subgroup, $h \circ h' = e$.

From Inverses are Unique, it follows that $h' = h^{-1}$.