Lefschetz numbers for sheaf-trivial proper surjections

Consider a fiber-preserving map of the total space of a fibration-like, proper surjection whose graded Leray sheaf is trivial. There is a Lefschetz number of this map, and a Lefschetz number for the map restricted to any fiber. We prove that the former is the product of the latter with the Euler characteristic of the base space. The result is based on a extension of the Hopf trace formula to finitary spectral sequences over fields.