Sum over k of r-k Choose m by s+k Choose n

Theorem
Let $m, n, r, s \in \Z_{\ge 0}$.

Then:
 * $\displaystyle \sum_{k \mathop = 0}^r \binom {r - k} m \binom {s + k} n = \binom {r + s + 1} {m + n + 1}$

where $\dbinom {r - k} m$ etc. are binomial coefficients.