Composition of Ring Homomorphisms is Ring Homomorphism

Theorem
Let:
 * $\left({R_1, +_1, \circ_1}\right)$
 * $\left({R_2, +_2, \circ_2}\right)$
 * $\left({R_3, +_3, \circ_3}\right)$

be rings.

Let:
 * $\phi: \left({R_1, +_1, \circ_1}\right) \to \left({R_2, +_2, \circ_2}\right)$
 * $\psi: \left({R_2, +_2, \circ_2}\right) \to \left({R_3, +_3, \circ_3}\right)$

be homomorphisms.

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