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 $$h' = h^{-1}$$.