The deficiency modules of Matlis duals of modules

Let (R,m) be a Noetherian local ring which is a quotient of a local Gorenstein ring. Let M be an R-module with a finitely generated submodule L such that M/L is Artinian. It is shown that the deficiency modules of Matlis dual of M is isomorphic to the deficiency modules of Matlis dual of M/L