Floor Function is Replicative

Theorem
The floor function is a replicative function in the sense that:


 * $\displaystyle \forall n \in \Z_{> 0}: \sum_{k \mathop = 0}^{n - 1} \left \lfloor{x + \frac k n}\right \rfloor = \left \lfloor{n x}\right \rfloor$