The last result we will need is the behavior of the universal equivariant Chern classes under Whitney sum. Let [mu]: BU x BU [right arrow] BU be the H-structure map which induces the Whitney sum on complex bundles.
III (Whitney sum) If [eta] = [[eta].sub.1] [??] [[eta].sub.2] then c([eta]) = c([[eta].sub.1])c([[eta].sub.2]), where c(-) is the total Chern class [[summation].sup.[infinity].sub.i=0][c.sub.i](-).
We are thus left with Axiom III about the total equivariant Chern class of a Whitney sum. As this class is defined by applying the section [??] to the usual Chern class, we have basically to check that the construction of the Whitney sum behaves well with respect to the [H.sup.*]-frame of a conjugation space.