Inclusion Mapping on Subgroup is Homomorphism

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

Let $\struct {H, \circ_{\restriction H} }$ be a subgroup of $G$.

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

Then $i$ is a group homomorphism.

Proof
Let $x, y \in H$.

From, $x \circ_{\restriction H} y \in H$.

Then:

Hence the result by definition of group homomorphism.