For a group $G$, if $K\unlhd H\leq G$ and $N\unlhd G$, then $NK\unlhd NH$? How come?
If $NK/N\unlhd NH/N$, then by correspondence theorem, $NK\unlhd NH$, but why is $NK/N\unlhd NH/N$? I know homomorphisms like the natural projection preserve normal subgroups, so wouldn't $\pi(K)\unlhd\pi(H)$, or $K/N\unlhd H/N$? Why is there the extra $N$ on top?