In Theorem 20.16 of his Introduction to smooth manifolds Lee proves that every complete Lie($G$)-action $\hat{\theta}:Lie(G)\to \Gamma (TM),$ where $G$ is a simply connected Lie group, is the infinitesimal generator of some smooth right $G$-action on $M.$
I‘ll quickly outline the idea of proof until the point where I am stuck:
Writing $\hat{X}$ for $\hat{\theta}(X)$ he defines for each $X\in Lie(G)$ a new vector field $\tilde{X}_{(g,p)}=(X_g,\hat{X}_p).$ He then goes on to define a distribution $D$ pointwise by declaring $D_{(g,p)}$ to be the set of all vectors $\tilde{X}_{(g,p)}$ as $X$ ranges over $Lie(G)$. He then shows that this distribution is involutive and so for each $p\in M$ he defines $S_p$ to be the leaf of the induced foliation containing $(e,p).$ Furthermore he sets $\Pi_p:=\pi_G|_{S_p}$ and wants to show that this is a smooth covering map. For each $q \in M$ such that $(g,q)\in \Pi_p^{-1}(g)$he defines a map $\sigma_q:V\to G\times M$ for a certain nbhd $V$ of $g$ and he showes that these are smooth local sections.
This is the point where I‘m missing something:
How do we know that there exist such a $q$? Or put differently, why is $\Pi_p:S_p\to G$ surjective? Except from this, everything else is clear to me.
I‘d be glad for explanations or hints in the right direction (maybe I have to read between the lines).
Thanks in advance