Product of r Choose m with m Choose k

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

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

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

Proof
Let $$r \in \Z$$.

Integral Index
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 $$r \in \R$$.