A short proof of a min-max relation for the bases packing of a matroid

Let E be a finite set, and M be a matroid defined on E. Given w is an element of R-+(E), we use the notations (w-maximum bases packing for the first one): lambda(w) = Max{Sigma(Bbasis) lambda(B) such that Sigma(B(sic)e) lambda(B) <= w(e) for any e is an element of E, and lambda(B) >= 0 for any basis B}, and w(l) = Min [GRAPHICS] such that U subset of E and r(U) <= r(E) - 1}. In this paper, we give a short proof for the known min-max relation lambda(w) = w(l). Moreover, we prove that the minimum w(l) can be restricted to single elements and semi locked subsets only. A subset L subset of E is semi locked in M if M*vertical bar(E\L) is closed and 2- connected, and min{r(L), r*(E\L)} >= 2. We deduce then a polynomial algorithm to compute w(l) in a large class of matroids by using a matroid oracle related to semi locked subsets.