Let $z \neq 0$ be a non-unit Gaussian integer. I wish to prove that the Gaussian integers in the fundamental parallelogram associated to $z$ are a complete residue system modulo $z$. I have succeeded in proving that the Gaussian integers inside the fundamental parallellogram are incongruent to each other, by showing that their difference cannot be an integral multiple of $z$. Now, I wish to show that for every Gaussian integer, there is a point in the fundamental parallelogram to which it is congruent modulo $z$. I have given the following argument, by feel it is "hand-wavey" and wish to make it more rigorous:
Any Gaussian integer $q$ can be located inside exactly one of the translates (or on its boundary) of the fundamental parallelogram; with the Division Algorithm, $$q = \beta z + r,$$ where $\beta z$ is the closest vertex in the translate and $r$ is the vector from this corner to $q$. There is a point $p$ located in the corresponding position inside the fundamental parallelogram, given by $$p = \beta' z + r,$$ where $\beta' z$ is the closest vertex in the fundamental parallelogram and $r$ is the same vector from this corner to $\mathcal{p}.$ Clearly, then, $$q \equiv_{z} p$$.
How can I formalise this proof?
