This paper derives state space error bounds for the solutions of reduced systems constructed using Proper Orthogonal Decomposition (POD) together with the Discrete Empirical Interpolation Method (DEIM) recently developed in [S. Chaturantabut and D.C. Sorensen, SISC 2010, pp. 2737-2764 ] for nonlinear dynamical systems. The resulting error bounds are shown to be proportional to the sums of the singular values corresponding to neglected POD basis vectors both in Galerkin projection of the reduced system and in the DEIM approximation of the nonlinear term. The analysis is particularly relevant to ODE systems arising from spatial discretizations of parabolic PDEs. The derivation clearly identifies where the parabolicity is crucial. It also shows exactly how the DEIM approximation error involving the nonlinear term comes into play.