I reading the proof of R. Schilling in his book : Brownian motion, introduction to stochastic process, and his proof is really unclear for me. I want to prove that if $u$ is bounded, then $$\mathbb E[u(B_{t+s})\mid \mathcal F_s]=\mathbb E[u(B_t)\mid B_s],$$ where $(\mathcal F_t)$ is an admissible filtration to $(B_t)$ (i.e. $B_t-B_s$ is independent of $\mathcal F_s$ for all $0\leq s\leq t$, and $(B_t)$ is a Brownian motion.
The proof goes as follow :
$$\mathbb E[u(B_{t+s})\mid \mathcal F_s]=\mathbb E[u(B_{t+s}-B_s+B_s)\mid \mathcal F_s]=\mathbb E[u(B_t+x)]|_{x=B_s}=\mathbb E[u(B_t)\mid B_s],$$ but I neither understand the notation $\mathbb E[u(B_t+x)]|_{x=B_s}$ nor the two last inequalities. I thought that $$\mathbb E[u(B_t+x)]|_{x=B_s}:=\mathbb E[u(B_t+B_s)\mid B_s=x],$$ but it doesn't really make sense. Any help is welcome.