Quotient Theorem for Group Homomorphisms/Corollary 2/Proof 2

Proof
Let $e$ be the identity element of $G$.

Let $\RR$ be the congruence relation defined by $N$ in $G$.

Let $\RR_\phi$ be the equivalence relation induced by $\phi$.

From Condition for Existence of Epimorphism from Quotient Structure to Epimorphic Image:
 * there exists an epimorphism $\psi$ from $G / N$ to $H$ which satisfies $\psi \circ q_N = \phi$


 * $\RR \subseteq \RR_\phi$
 * $\RR \subseteq \RR_\phi$

It remains to be shown that:
 * $\RR \subseteq \RR_\phi$


 * $N \subseteq K$
 * $N \subseteq K$