Product of r Choose m with m Choose k

Theorem
Let $r \in \R, m \in \Z, k \in \Z$.

Then:
 * $\dbinom r m \dbinom m k = \dbinom r k \dbinom {r - k} {m - k}$

where $\dbinom r m$ is a binomial coefficient.

Integral Index
Let $r \in \Z$.

Then:

Real Index
Both sides of the above equation are polynomials in $r$.

Since these polynomials agree for all $r \in \Z$, they must agree for all $r \in \R$.