Let $G:\mathscr R^n\to\mathscr R^m$ be a $\mathscr C^1$ mapping, where $k = n-m>0$. If $M$ is the set of all those points $\mathbf x\in G^{-1}(\mathbf 0)$ for which the derivative matrix $G'(\mathbf x)$ has rank $m$, then $M$ is a smooth $k$-manifold.
What I took from this is the following:
If $M=\{x: G'(x) \text{ has rank } m \text{ and } x\in G^{-1}(0) \}$, then $M$ is a smooth $k$-manifold.
But a fellow student claims it should be:
If $G'(x)$ has rank $m$ for all $x\in G^{-1}(0)$ only then $M=\{x:x\in G^{-1}(0)\}$ is a smooth $k$-manifold
Which one is correct?