Quotient of Modulo Operation with 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 \left \lfloor {\dfrac x y}\right \rfloor & : y \ne 0 \\ x & : y = 0 \end{cases}$ where $\left \lfloor {\dfrac x y}\right \rfloor$ denotes the floor of $\dfrac x y$.

Let $y \ne 0$.

Then:


 * $0 \le \dfrac x y - \left \lfloor {\dfrac x y}\right \rfloor = \dfrac {x \bmod y} y < 1$

Proof
From Real Number minus Floor:
 * $\dfrac x y - \left \lfloor {\dfrac x y}\right \rfloor \in \left[{0 \,.\,.\, 1}\right)$

Thus by definition of half-open real interval:
 * $0 \le \dfrac x y - \left \lfloor {\dfrac x y}\right \rfloor < 1$

Then:

Hence the result.