Composite of Group Homomorphisms is Homomorphism

Theorem
Let:
 * $\struct {G_1, \circ}$
 * $\struct {G_2, *}$
 * $\struct {G_3, \oplus}$

be groups.

Let:
 * $\phi: \struct {G_1, \circ} \to \struct {G_2, *}$
 * $\psi: \struct {G_2, *} \to \struct {G_3, \oplus}$

be homomorphisms.

Then the composite of $\phi$ and $\psi$ is also a homomorphism.