The complete Nevanlinna-Pick property (CNP) of a reproducing kernel Hilbert space is related to the solution of Nevanlinna-Pick interpolation problems, and has been studied extensively in the literature. One of the fundamental problem is to characterize: which reproducing kernel Hilbert spaces have the CNP? We will talk about the CNP of the sub-Hardy/Bergman space on polydisk, and see that when the kernel $k$ tensor product or Schur product with an additional factor, the CNP will diminish.