On $p$-supersoluble residual of a product of normal $p$-supersoluble subgroups

Let a finite group $G$ be a product of two normal $p$-supersoluble subgroups. We prove that the $p$-supersoluble residual of $G$ coincides with the $p$-nilpotent residual of the commutator subgroup of $G$. Hence it follows that the supersoluble residual of a product of normal supersoluble subgroups coincides with the nilpotent residual of the commutator subgroup.