$(2\Bbb{Z} + 1)\cup 3\Bbb{Z} = 2\Bbb{Z} \cup 3\Bbb{Z} + 3$

Proof:

$$
\begin{align\*}
2\Bbb{Z} &= \bullet \circ \bullet \circ \bullet \circ \bullet \circ \dots \\
3\Bbb{Z} &= \bullet \circ \circ \bullet \circ \circ \bullet \circ \dots \\
2\Bbb{Z}+1 &= \circ \bullet \circ \bullet \circ \bullet \circ \bullet \dots \\
2\Bbb{Z} \cup 3\Bbb{Z} &= \bullet \circ \bullet \bullet \bullet \circ \bullet \circ \dots \\
(2\Bbb{Z} + 1)\cup 3\Bbb{Z} &= \bullet \bullet \circ \bullet \circ\bullet\bullet\bullet \dots
\end{align\*}
$$
Shift the last one by $\pm 3$. Conclude. That's a visual proof. But what is the underlying algebra?