Rolle's theorem for polynomials

Setup. Let $x_1<x_2<\cdots<x_n$ be the roots of $P$. On each interval $[x_i,x_{i+1}]$, $P$ is continuous on the closed interval, differentiable on the open interval, and $P(x_i)=P(x_{i+1})=0$.

Rolle. By Rolle's theorem, there exists $c_i\in]x_i,x_{i+1}[$ with $P'(c_i)=0$.

Distinctness. The $c_i$ are in disjoint intervals, hence distinct.

Conclusion. $P'$ has at least $n-1$ distinct real roots. This is exactly the statement.