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

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:
 * $0 = b \paren {q_1 - q_2} + \paren {r_1 - r_2}$

that $q_1 \ne q_2$.

, suppose 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.