We prove that virtual friction control can stabilize a power grid containing several virtual synchronous machines (VSMs), connecting line impedances and loads. Virtual friction is a torque added to the swing equation of each VSM, proportional to the deviation of its frequency from the overall center of inertia (COI) frequency. Our analysis is based on the network reduced power system (NRPS) model. We support our results with simulations for a two-area network of four VSMs, looking at the transients induced by a change of tie-line impedance and an asymmetric load change. We compare the results for the NRPS model with the corresponding results using detailed models of synchronverters and line impedances. We find that virtual friction has a strong stabilizing effect both for the NRPS model and for the full grid model.