Sum over k of r-tk Choose k by s-t(n-k) Choose n-k by r over r-tk/Proof 1/Lemma

Theorem
Let this hold for $\left({r, s, t, n}\right)$:
 * $\displaystyle \sum_{k \mathop \ge 0} \binom {r - t k} k \binom {s - t \left({n - k}\right)} {n - k} \frac r {r - t k} = \binom {r + s - t n} n$

and also for $\left({r, s - t, t, n - 1}\right)$.

Then it also holds for $\left({r, s + 1, t, n}\right)$.

Proof
Evaluating the equation for $\left({r, s - t, t, n - 1}\right)$:

Adding the equation in $\left({r, s, t, n}\right)$:

Hence the equation holds for $\left({r, s + 1, t, n}\right)$