Let X be a semialgebraic set in Rⁿ defined by a Boolean combination of atomic formulae of the kind h * 0 where * \in { >, \ge, = }, deg(h) < d,="" and="" the="" number="" of="" distinct="" polynomials="" h="" is="" k.="" we="" prove="" that="" the="" sum="" of="" betti="" numbers="" of="" x="" is="" less="" than="">²d)ⁿ.