Given a flow $\Phi_i(t,p)$ defined on $A_i=\mathbb{R}\times E$, and $f$ is a proper map from $E$ to $\mathbb{R}^n$, s.t. $f\Phi_i(t,p)=f(q)+e_it$. And define $J_q$ by $A_i=\cup_{q\in E}J_q\times\{q\}$. Now the following proof shows $\Phi_i(t,p)$ is global.
The Lemma 7.3.6 is here:
My question is we don't know if $J_q$ is bounded, how can we apply what the author said about bounded open interval and the lemma 7.3.6 to give the contradiction?