ve assertion, we observe that the P_i form a covering of P.
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| Furthermore, the manifold structure on P_i |^| P_j induced by that of P_i or P_j
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| must be the same, because it is the unique fiber product structure over V_i |^| V_j,
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| for the maps f_ij and g_ij (defined on f^(-1)(V_i |^| V_j) and g^(-1)(V_i |^| V_j),
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| respectively).  Thus we can give P a manifold structure, in such a way that the
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| two projections into X and Y are morphisms, and make P into a fiber product
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| of f and g.
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| We sha