$\newcommand{\dplus}{\mathbin{\dot{+}}}$ Denote the dyadic addition by $\dplus$. For any $x,y\in[0,1)$ with $$ x=\sum\limits_{k=0}^\infty\frac{x_k}{2^{k+1}} \quad\mbox{and}\quad y=\sum\limits_{k=0}^\infty\frac{y_k}{2^{k+1}}, $$ where, for any $k\in\mathbb{Z}_+$, $\,x_k,y_k\in\{0,1\}$, define $$ x \dplus y := \sum\limits_{k=0}^\infty \frac{|x_k-y_k|}{2^{k+1}} $$ and for any measurable subset $E$ of $[0,1)$, define $$ y \dplus E := \{y \dplus x:\, x\in E\}. $$ Let $I=\left[\frac{m}{2^k}, \frac{m+1}{2^k}\right)$ be a dyadic interval, where $k\in\mathbb{Z}_+,\, 0\leq m\leq 2^k-1$.
How to prove that, for any $l<k$ and $l\in\mathbb{Z}_+$, $$ I \dplus 2^{-l-1} $$ is a dyadic interval ?