Can anyone help me prove the following:

Let f: R^{n+k}\to R^{n} be of class C^{r}. Let M be the set of all x such that f(x) = 0. Assume that M is non-empty and that Df(x) has rank n for x in M.
Then M is a k-manifold without boundary in R^{n+k}. Furthermore, if N is the set of all x for which f_{1}(x) = ... = f_{n-1}(x) and f_{n}(x) \ge 0, and if the matrix \partial (f_{1},...,f_{n-1})/\partial x has rank n-1 at each point of N, then N is a k+1 manifold, and \partial N = M