Exercise 9, § 2

Let $G$ be a quasi-topological group, and let $H$ be a closed normal subgroup of $G$ which contains the commutator subgroup of $G$. Show that if the component $K$ of $e$ in $H$ is solvable, then so is the component $L$ of $e$ in $G$ [show by use of Exercise 7 a) that $K$ contains the commutator subgroup of $L$].