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A method is proposed to infer Lyapunov and asymptotic stability properties for switching systems, under arbitrary continuous-state feedback. Continuous-time systems which are dissipative in the multiple-storage function sense are considered. A partition of the state space, induced by the cross-supply rates and the feedback function, is used to derive conditions for stability. It is argued that the conditions proposed here are more straightforward to check, when compared to those proposed by other approaches in the literature. Some numerical examples are offered to illustrate this point.