I am studying a proof of a theorem that states that a subgroup $N$ of a group $G$ is normal in $G$ if and only if $(xA)(yA) = (xy)A$ for all $x,y \in G$. The author goes through a fairly involved element-chase to prove necessity. However, it seems clear that if $N$ is normal in $G$ that $ (Nx)(Ny) = (Nx)(yN) = N(xy)N = (xy)NN = (xy)N $ where only normality and the nature of the "product" $AB = \{ab | a \in A, b \in B\}$ is used.
Does my argument work or am I overlooking a detail?