If $H$ is a subloop of a finte loop $L$ and $N$ is a normal subloop of $L$, then $HN$ is a subloop of $L$.

Question

To prove this is a subloop, I have to show that for $x, y \in HN$, the following are also in $HN$: (a) $xy$, (b) $L^{-1}_x(y)$ and (c) $R^{-1}_{x}(y)$.

Here $L_x(y) = xy$ and $R_x(y)=yx$. We have to show that inverses of these translation maps are in $HN$.

The first part (a) that the product is inside $HN$ is easy. I need help with both/one of the others.