Let $\displaystyle x=(A_x,B_x), y=(A_y,B_y), x,y\in\mathbb{R}$ be two Dedekind cuts. Is it true that $\displaystyle (A_x+A_y)\cup(B_x+B_y)=\mathbb{Q}$? where $\displaystyle A_x+A_y=\{r+s|r\in A_x\;{\rm{and}}\;s\in A_y\}$, the meaning of $\displaystyle B_x+B_y$ is similar.

I met with this problem while trying to prove the equivalence of two formulations of addition of real numbers, therefore, to avoid iterative proof please restrict addition to rationals only. Thanks!