The Lefschetz Theorem on Hyperplane Sections

is bijective for i < n - 1 and surjective for i = n - 1. Several proofs of this theorem are to be found in the literature (see 5 for an account of the problem). Recently Thom has given a proof (unpublished) which, as far as we know, is the first to use Morse's theory of critical points. We present in ? 3, in a slightly more general setting, an alternate proof inspired by Thom's discovery. Our statement is given in the equivalent language of cohomology. The proof is derived from a theorem on Stein manifolds which is presented in ? 2. Some standard properties of the distance function which we require are assembled in ? 1 for the sake of completeness.