Let $a \in \mathbb{Z}$. Then there is a unique integer $r \in \{0,1, \dotsc, n-1\}$ such that $a \equiv r \, \mathrm{mod} \, n$.
My question is with regards the unique existence of the nonnegative integer $r$. In the proof in the text, the author provided separate cases typically associated with unique existence proofs, showing uniqueness and existence separately.
My question is, since we can use the division algorithm upon dividing $a$ by $n$, we immediately obtain unique integers $q$ and $r \in \{0,1,\dotsc,n-1\}$ such that
$$a-r = nq$$
which implies that
$$a \equiv r \, \mathrm{mod} \, n$$
so I don't see how it is necessary to treat uniqueness and existence separately.