Let $R=[a_1, b_1] \times ... \times [a_n , b_n]$ be a rectangle in $\mathbb{R} ^n$. Then:
$R_1 , R_2$ are rectangles in $\mathbb{R} ^n$ such that $R=R_1 \cup R_2$ and $int(R_1)\cap int(R_2) = \emptyset $ if and only if there exists $i\in \{ 1,...,n\}$ and $c \in (a_i , b_i ) $ such that $R_1=[a_1, b_1] \times ... \times [a_i, c_i] \times ... \times [a_n , b_n]$ and $R_2=[a_1, b_1] \times ... \times [c_i, b_i] \times ... \times [a_n , b_n]$.
The converse is clear to me. Now, the other implication is intuitively clear (at least in $\mathbb{R} ^2 $ and $\mathbb{R} ^3 $), but I don't know how to write this formally. Any hint would be really appreciated. Thank you so much!