Assume $\left\{ W_{t},\mathscr{F}_{t}:0\le t<\infty\right\} $ is a one-dimensional Brownian motion. How can we show that $$P\left\{ W_{t+s}\le a,\max_{0\le u\le s}W_{t+u}\le b|\mathscr{F}_{t}\right\} = P\left\{ W_{t+s}\le a,\max_{0\le u\le s}W_{t+u}\le b|W_{t}\right\} $$
The equation is true in Karatzas/Shreve in the book "Brownian Motion and Stochastic Calculus". I know that Brownian motion is a Markov process, but I do not know how to prove this equation. Can anyone help me?