Sum over j of Function of Floor of mj over n/Corollary

Corollary to Sum over $j$ of $f \left({\left \lfloor{\frac {m j} n}\right \rfloor}\right)$

 * $\displaystyle \sum_{0 \mathop \le j \mathop < n} \dbinom {\left \lfloor{m j / n}\right \rfloor + 1} k + \sum_{0 \mathop \le j \mathop < m} \left \lceil{\dfrac {j n} m}\right \rceil \binom j {k - 1} = n \binom m k$

Proof
Hence: