Proof: Let us first consider the case "n" = 1. In this case, U(1) is the circle S1 and the universal bundle is S∞ → CP∞. It is well known that the cohomology of CP"k" is isomorphic to formula_17, where "c"1 is the Euler class of the U(1)-bundle S2"k"+1 → CP"k", and that the injections CP"k" → CP"k"+1, for "k" ∈ N*, are compatible with these presentations of the cohomology of the projective spaces. This proves the Proposition for "n" = 1.