Preimage of Normal Subgroup of Quotient Group under Quotient Epimorphism is Normal/Proof 2

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

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

Let $\SS$ be the congruence relation defined by $K$ in $G / H$.

Let $\TT$ be the relation on $G$ defined as:
 * $\forall x, y \in G: x \mathrel \TT y \iff x H \mathrel \SS y H$

From Equivalence Relation induced by Congruence Relation on Quotient Structure is Congruence:
 * $\TT$ is a congruence relation on $G$

Hence from Congruence Relation on Group induces Normal Subgroup:
 * the equivalence class under $\TT$ of $e$, that is $\eqclass e \TT$, is a normal subgroup of $G$.

Let $L := \eqclass e \TT$.

Then we have:

and so $L = q_H^{-1} K$ is a normal subgroup of $G$.