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This note presents new LMI-based stability criteria for the discrete-time multivariable Lur'e system with nonlinearities that are monotonic, sector- and slope-restricted. Corresponding Lur'e-Lyapunov functions are constructed for such a system. The new criteria are expressed in a reasonably general form that can be applied to both non-diagonal and diagonal nonlinearities. We explicitly compare the new approach to the existing LMI-based Popov-like criteria in the literature, and express the results in terms of an Integral Quadratic Constraint (IQC). The applications of the new criteria to some control problems and strategies are briefly discussed. Numerical examples are included to show their performance, and they are shown to be less conservative than the previous results.