In this paper, new properties of the cascade between a multi-input multi-output linear time-invariant system, referred to in the following as the original system, and an inner system are dealt with. In particular, attention is devoted to the relationship between the Hankel singular values and characteristic values of the cascade system and those of the original one, proving that, if the inner system has gain greater than or equal to one, then the first $n$ Hankel singular values (characteristic values) of the cascade system are greater than or equal to those of the original system. The results are then applied to derive a new property of minimum phase systems.