I'm following the book "Elementary Introduction To The Theory Of Pseudodifferential Operators" by X. S. Raymond and the Joshi Lectures Notes - https://arxiv.org/pdf/math/9906155.pdf - to prove the Propagation of Singularities Theorem.

Since $WF(u)$ is a closed set and a bicharacteristic curve $\gamma$ is a connected set, the only fact to prove is that $WF(u) \cap \gamma$ is an open subset of $\gamma$. Thus, given a $(x_0,\xi_0) \in WF(u) \subset \operatorname{Char} a$, we want to prove that the bicharacteristic curve $(x(t),\xi(t)) $starting at this point is locally contained in $WF(u)$. We will proceed by contradiction, assuming that this is not true.

I could not see where the contradiction hypothesis -there is a point $(x(s),\xi(s)) \not\in WF(u)$ for an $s$ in the domain of definition of $\gamma$ - is used in the proof (Raymond, p. 83).

Can anybody help me?

Update: I think I understood, it remains to show two things: $\operatorname{supp} c(s) \subset \operatorname{supp}c_0 \subset e^{sH_q}\operatorname{supp}c_0$ and $w(s)=c(s,c,D)v=\chi \in C_0^{\infty}$.

Then, by a reversion in the time we will have that $\tilde{w}(t)=w(t+s)$ is solution of the problem $\partial \tilde{w}(t)-ib(x,D)\tilde{w}(t)=g(t+s)$ with $\tilde{w}(0)=\chi \in C_{0}^{\infty}$. Hence, $\tilde{w} \in \bigcap_{k} C^{0}(I:H^{k})$. In particular, $\tilde{w}(-s)=w(0)=c(x,D)v \in C^\infty$, which proves that $(x_0,\xi_0) \not\in WF(u),$ since $c_0(x_0,\xi_0)\neq 0.$

P.S: I have posted this in Math Stack Exchange but didn't get a reply and so I have posted it here. The page mentioned is in the link below.