Douglas' Factorization Theorem and Atomic System in Hilbert Pro-$C^{\ast}$-Modules

In the present paper, we introduce  the generalized inverse operators, which have an exciting role in operator theory. We establish Douglas' factorization theorem type  for  the Hilbert pro-$C^{\ast}$-module.We introduce the notion of atomic system and $K$-frame in the Hilbert pro-$C^{\ast}$-module and study their relationship. We also demonstrate some properties of the $K$-frame by using Douglas' factorization theorem.Finally  we demonstrate that the sum of two $K$-frames in a Hilbert pro-$C^{\ast}$-module with certain conditions is once again a $K$-frame.