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The current analysis of atmospheric neutrinos by the Super-Kamiokande Collaboration is based only on fully-contained one-ring events and partially contained events. We show that the up-down ratio of fully-contained, inclusive, multi-ring events gives an independent test of the atmospheric neutrino anomaly, without the need for particle identification. Moreover, this class of events is rich in neutral current events and hence gives crucial information for discriminating between oscillations of \nu_\mu into \nu_{e, \tau} and \nu_s.

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