Parusi\'nski's "Key Lemma" via algebraic geometry

The following ``Key Lemma'' plays an important role in Parusinski's work on the existence of Lipschitz stratifications in the class of semianalytic sets: For any positive integer n, there is a finite set of homogeneous symmetric polynomials W_1,...,W_N in Z[x_1,...,x_n] and a constant M >0 such that |dx_i/x_i| \le M \max_{j = 1,..., N} |dW_j/W_j| as densely defined functions on the tangent bundle of C^n. We give a new algebro-geometric proof of this result.