proof of uniqueness of Lagrange Interpolation formula

Existence is clear from the construction, the uniqueness is proved by assuming there are two different polynomials $p(x)$ and $q(x)$ that interpolate the points. Then $r(x)=p(x)-q(x)$ has $n$ zeros, $x_1,\mathrm{\dots },x_n$ and there is a point $x_e$ such that $r(x_e)\ne 0$. $r(x)$ is non-constant with degree $\mathrm{deg}(r(x))\le n-1$ and has more than $n-1$ solutions, which is a contradiction. Thus there can only be one polynomial.