Direct image for multiplicative and relative K-theories from transgression of the families index theorem, part 3

This is the final part of the work started in math.DG/0611281 and math.DG/0703916. Here the question of double fibration ois adressed both for relative k-theory and free multiplicative K-theory. In the case of relative and ``nonfree’’ multiplicative K-theory, the direct image is proved to be functorial for double submersions.