## Proof of the cell decomposition theorem

Let $\M$ be an o-minimal expansion of a dense linear order $(M,\lt)$, and let $n \in \NN$. We now prove the cell decomposition theorem by induction on $n$, assuming that $n \ge 2$ and that (I)$_{n-1}$ and (II)$_{n-1}$ hold. Proof of (I)$_n$. As in the proof of the decomposition theorem for planar sets, we may…