We first notice that in the above statement, $U$ can be replaced by any smaller open set that meets all the log canonical centers: the closure of the restriction of $\sum^{j}_{i=\ell+1}\Delta_{i}|_{U}$ to $U\cap D$ will yield the same divisor on $D$.

Then by induction, we only need to treat the case where $D$ is a component of $\Delta$, say $\Delta_{1}$. If we take a log resolution $f:Y\to(Z,\Delta)$ and let $D^{\prime}=\Delta^{\prime}_{1}$ be the birational transform of $D$, then $\Delta_{D}$ can be computed as follows: if we write $f^{*}(K_{Z}+\Delta)|_{D^{\prime}}=K_{D^{\prime}}+\Delta_{D^{\prime}}$ then $\Delta_{D}=(f|_{D^{\prime}})_{*}(\Delta_{D^{\prime}})$. In particular, as $K_{Z}+\Delta$ is Cartier, we know that $\Delta_{D}$ is a integral divisor. Since it is effective, and all the components of $\Delta_{D}$ are log canonical centers of $(X,\Delta)$, we see that $\Delta_{D}$ must be equal to the closure of the restriction of $\sum^{j}_{i=2}\Delta_{i}|_{U}$. ∎