Measure of inverse image of points by an analytic mapping

Question

How can one prove the following statement: For any analytic mapping from a connected analytic manifold $M$ to an analytic manifold $N$, the inverse image of a point in $N$ is either the whole of $M$ or has measure zero in $M$. The hint is taking $M$ to be an open subset of $\mathbb{R}^m$ and using induction on $m$, and Fubini’s theorem. But I can't fill in the details.