Division Theorem/Positive Divisor/Existence/Proof 2

Theorem
For every pair of integers $a, b$ where $b > 0$, there exist integers $q, r$ such that $a = q b + r$ and $0 \le r < b$:


 * $\forall a, b \in \Z, b > 0: \exists q, r \in \Z: a = q b + r, 0 \le r < b$

Proof
Let:
 * $q = \left \lfloor{\dfrac a b}\right \rfloor, t = \dfrac a b - \left \lfloor{\dfrac a b}\right \rfloor$

where $\left \lfloor{\cdot}\right \rfloor$ denotes the floor function.

Thus $q \in \Z$ and $t \in \left[{0 \,.\,.\, 1}\right)$.

So:
 * $\dfrac a b = q + t$

and so:
 * $(1): \quad a = q b + r$

where $r = t d$.

Since $a, q, b \in \Z$, it follows from $(1)$ that:
 * $r = a - q b$

and so $r \in \Z$ also.

Since $0 \le t < 1$ and $b > 0$, it follows that:
 * $0 \le t b < b$

that is:
 * $0 \le r < b$