= Solution
For $d=p-q$, all $d(x_i)$ and $d'(x_i)$ vanish. Between each consecutive pair of nodes, <Rolle's theorem> gives one additional zero of $d'$. Along with the $n+1$ nodal zeros, this supplies $2n+1$ distinct zeros on the closed interval. Since $d'$ is a <polynomial> of degree at most $2n$, it must be the zero <polynomial>. Thus $d$ is constant, and $d(x_0)=0$ makes it zero. Hence \b[the interpolating <polynomial> is unique].
The direct zero count includes the two endpoints; its guaranteed interior count before the degree argument is $2n-1$. After $d'=0$ is proved, $d'$ vanishes at every point of the open interval, so the stronger interior-zero assertion in the source follows as well. Another immediate uniqueness proof is that $d$ has a double zero at every node, so is divisible by $\prod_i(x-x_i)^2$ of degree $2n+2$, although its own degree is at most $2n+1$.
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