Composite of Epimorphisms is Epimorphism

Theorem
Let:
 * $\left({S_1, \circ_1, \circ_2, \ldots, \circ_n}\right)$
 * $\left({S_2, *_1, *_2, \ldots, *_n}\right)$
 * $\left({S_3, \oplus_1, \oplus_2, \ldots, \oplus_n}\right)$

be algebraic structures.

Let:
 * $\phi: \left({S_1, \circ_1, \circ_2, \ldots, \circ_n}\right) \to \left({S_2, *_1, *_2, \ldots, *_n}\right)$
 * $\psi: \left({S_2, *_1, *_2, \ldots, *_n}\right) \to \left({S_3, \oplus_1, \oplus_2, \ldots, \oplus_n}\right)$

be epimorphisms.

Then the composite of $\phi$ and $\psi$ is also an epimorphism.

Proof
From Composite of Homomorphisms on Algebraic Structure is Homomorphism, $\psi \circ \phi$ is a homomorphism.

From Composite of Surjections is Surjection, $\psi \circ \phi$ is a surjection.

An epimorphism is a surjective homomorphism.

Hence $\psi \circ \phi$ is an epimorphism.