Range of Modulo Operation for Negative Modulus

Theorem
Let $x, y \in \R$ be real numbers.

Let $x \bmod y$ denote the modulo operation:
 * $x \bmod y := \begin{cases}

x - y \floor {\dfrac x y} & : y \ne 0 \\ x & : y = 0 \end{cases}$ where $\floor {\dfrac x y}$ denotes the floor of $\dfrac x y$.

Let $y < 0$.

Then:


 * $0 \ge x \bmod y > y$

Proof
Hence the result.