Given $$x_1>0,\quad x_2>0,\quad x_3>0,\quad x_4>0,\quad\textrm{and}\quad m>0,$$ could we claim that the system
$$\begin{cases} [x_1+x_2]_m = 0 \\ [x_1+x_4]_m = 0 \\ [x_3+x_2]_m = 0 \\ [x_3+x_4]_m = 0 \end{cases}$$
has a solution only when
$$ [x_1]_m = [x_2]_m = [x_3]_m = [x_4]_m = 0 $$
?
