uniqueness of division algorithm in Euclidean domain

Theorem.  Let $a,b$ be non-zero elements of a Euclidean domain $D$ with the Euclidean valuation $\nu$.  The incomplete quotient $q$ and the remainder $r$ of the division algorithm

are unique if and only if

$\nu (a+b)\leqq \max \{\nu (a),\nu (b)\}.$(1)

Proof.  Assume first (1) for the elements $a,b$ of $D$.  If we had

and  $r^{\prime }\ne r$,  $q^{\prime }\ne q$,  then the properties of the Euclidean valuation (http://planetmath.org/EuclideanValuation) and the assumption yield the of inequalities

which is impossible.  We must infer that  $r^{\prime }-r=0$  or  $q^{\prime }-q=0$.  But these two conditions are equivalent (http://planetmath.org/Equivalent3).  Thus the division algorithm is unique.

Conversely, assume that (1) is not true for non-zero elements $a,b$ of $D$, i.e.

$$\nu (a+b)>\max \{\nu (a),\nu (b)\}.$$

Then we obtain two repsesentations

$$b=0(a+b)+b=1(a+b)-a$$

where    and  .  Thus the incomplete quotient and the remainder are not unique.