Division Theorem/Positive Divisor/Positive Dividend

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


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

In the above equation:
 * $a$ is the dividend
 * $b$ is the divisor
 * $q$ is the quotient
 * $r$ is the principal remainder, or, more usually, just the remainder.

Proof
This result can be split into two parts: