Join lemma for normal subgroup of subgroup with normal subgroup of whole group

Statement with symbols
If $$H \underline{\triangleleft} K \le G$$ and $$L \underline{\triangleleft} G$$, then $$\langle H, L \rangle \underline{\triangleleft} \langle K, L \rangle$$.

Related facts

 * Normality is strongly join-closed
 * Normality is upper join-closed

Applications

 * Join of subnormal subgroup and normal subgroup is subnormal