In page 307, Section 5.7, Chapter 5 of the book "mathematical methods for financial markets" by Jeanblanc, Yor and Chesney, Lemma 5.7.1 is given as follows:
Lemma 5.7.1.1 Let $W$ be a Brownian motion, $L$ its local time at level $0$ and $\tau_{l}=\inf\{ t:L_{t}\geq l\}$. Then
$$(W_{u}, u\leq\tau_{l}|\tau_{l}=t)\overset{law}{=}(W_{u}, u\leq t|L_{t}=l, W_{t}=0)\quad(1)$$ As a consequence, $$(W_{\tau_{l}-u}, u\leq\tau_{l})\overset{law}{=}(W_{u}, u\leq \tau_{l})$$ Its proof begins as assuming (1), then---.
My question is how to prove (1). There is no proof in their proof. Is it trivial? I am confused the condition $W_{t}=0$ in the right side of (1). Any references are very appreciated.