Quotient Theorem for Group Homomorphisms/Corollary 2/Proof 1

Proof
From Quotient Theorem for Group Homomorphisms: Corollary 1:


 * $N \subseteq K$


 * there exists a group homomorphism $\psi: G / N \to H$ such that $\phi = \psi \circ q_N$
 * there exists a group homomorphism $\psi: G / N \to H$ such that $\phi = \psi \circ q_N$

From Surjection if Composite is Surjection, it follows that the group homomorphism $\psi$ is a surjection.

Hence by definition, $\psi$ is an epimorphism.