Division Theorem/Positive Divisor/Positive Dividend/Uniqueness/Proof 1

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


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

Proof
It is given by Division Theorem: Positive Divisor: Positive Dividend: Existence that such $q$ and $r$ exist.

Suppose $q_1, r_1$ and $q_2, r_2$ are two pairs of $q, r$ that satisfy $a = q b + r, 0 \le r < b$.

That is:

This gives:
 * $0 = b \left({q_1 - q_2}\right) + \left({r_1 - r_2}\right)$

Aiming for a contradiction, suppose $q_1 \ne q_2$.

Suppose WLOG that $q_1 > q_2$.

Then:

This contradicts the assumption that $r_2 < b$.

A similar contradiction follows from the assumption that $q_1 < q_2$.

Therefore $q_1 = q_2$ and so $r_1 = r_2$.

Thus it follows that $q$ and $r$ are unique.