Summation over k of Floor of x plus k over y

Theorem
Let $x, y \in \R$ such that $y > 0$.

Then:


 * $\displaystyle \sum_{0 \mathop \le k \mathop < y} \left \lfloor{x + \dfrac k y}\right \rfloor = \left \lfloor{x y + \left \lfloor{x + 1}\right \rfloor \left({\left \lceil{y}\right \rceil - y}\right)}\right \rfloor$