Generic fiber rings of mixed power series/polynomial rings

Let K be a field, m and n positive integers, and X={x1,…,xn}, and Y={y1,…,ym} sets of independent variables over K. Let A be the localized polynomial ring K[X](X). We prove that every prime ideal P in that is maximal with respect to P∩A=(0) has height n−1. We consider the mixed power series/polynomial rings B:=K〚X〛[Y](X,Y) and C:=K[Y](Y)〚X〛. For each prime ideal P of that is maximal with respect to either P∩B=(0) or P∩C=(0), we prove that P has height n+m−2. We also prove each prime ideal P of K〚X,Y〛 that is maximal with respect to P∩K〚X〛=(0) is of height either m or n+m−2.