Math 23a: Proof 9.1 by Raviv Murciano-Goroff:
Let W be an open subset of Rn+m, and let F:U→Rn be a C1 mapping such that F(c)=0. Assume that [DF(c)] is onto.
Prove that the n + m variables can be ordered so that the first n columns of [DF(c)] are linearly independent, and that [DF(c)] = [A|B] where A is an invertible n × n matrix.