Corollary

Assume that $A$ is an integral domain and let $f, g \in A[X]$, of degrees $\leq n$. If there exist $n + 1$ pairwise distinct elements $x_1, \ldots, x_{n+1}$ of $A$ such that $f(x_i) = g(x_i)$ for $1 \leq i \leq n + 1$, then $f = g$.

It suffices to apply Th. 1 to $f - g$.