Inclusion Mapping on Subgroup is Homomorphism

Theorem
Let $\left({G, \circ}\right)$ be a group.

Let $\left({H, \circ_{\restriction H}}\right)$ be a subgroup of $G$.

Let $\iota: H \to G$ be the inclusion mapping from $H$ to $G$.

Then $\iota$ is a group homomorphism.

Proof
Let $x, y \in H$.

Then: