Let $r \in \mathbb{N}$. Observe the equivalence relation $\sim \subseteq \mathbb{Z} \times \mathbb{Z}$ defined as $x \sim y :\Leftrightarrow (\exists k \in \mathbb{Z})(y=x+kr)$. Show that an operation '+' on $\mathbb{Z}/\sim$ given by $[x]_\sim + [y]_\sim := [x+y]_\sim$ is well-defined.
My attempt at a proof:
Let x\sim x', y\sim y', then we know that (\exists k_1 \in \mathbb{Z})(x=x'+k_1r) and (\exists k_2 \in \mathbb{Z})(y=y'+k_2r). Adding $x$ and $y$, we see that x+y=x'+y'+(k_1+k_2)r. Because $\mathbb{Z}$ is closed under addition, $k_1+k_2 \in \mathbb{Z}$, and eventually x+y\sim x'+y'.
Unfortunately, I'm not even sure whether I'm on the right track here. As far as I can see, I've shown that adding two elements of (different) equivalence classes yields an element, whose equivalence class is in $\mathbb{Z}/\sim$. Is that sufficient to show that '+' is well-defined?