Fibers of a locally trivial fibration are diffeomorphic There is an "immediate" corollary in this paper that is not so immediate for me :
https://people.math.osu.edu/george.924/Ehresmann%20Theorem
This paper proves Ehresmann’s Theorem which states that every proper submersion $F:M\rightarrow N$ is locally trivial. In other words, to every point $p\in N$ corresponds a neighborhood $U$ and a diffeomorphism $\Psi:F^{-1}(U)\rightarrow U\times F^{-1}(p)$ such that : $F|_{F^{-1}(U)}=\pi\circ\Psi$, where $\pi:U\times F^{-1}(p)\rightarrow U$ is the natural projection.
What I am not able to prove is that if $F$ is a proper submersion, then all of its fibers are diffeomorphic using the theorem stated above. I think the reason why I can't seem to get a hold on that is because the theorem gives a local result, so I can't see how the theorem would help if I pick two arbitrary points in $N$ and and if I want to prove that their preimages are diffeomorphic.
 A: Let $p\in N$, there exists a neighborhood $U$ and a diffeomorphism $\Psi:F^{-1}(U)\rightarrow U\times F^{-1}(p)$ such that : $F|_{F^{-1}(U)}=\pi\circ\Psi$. First, let's consider $q\in U$. Define $\pi':U\times F^{-1}(p)\rightarrow F^{-1}(p)$ the other projection. And define also $\Theta:=\pi'\circ\Psi:F^{-1}(q)\rightarrow F^{-1}(p)$. $\Theta$ is a bijection whose inverse is given explicitely by $\Theta^{-1}:F^{-1}(p)\rightarrow F^{-1}(q),y\mapsto\Psi^{-1}(q,y)$. It is not hard to see that $\Theta$ and $\Theta^{-1}$ are both smooth maps.
Now let's consider a point $z\notin U$, so there exists a neighborhood $V$ and a diffeomorphism $\Psi':F^{-1}(V)\rightarrow V\times F^{-1}(z)$ such that : $F|_{F^{-1}(V)}=\pi\circ\Psi$. If suppose that there exists a point $q\in U\cap V$, then there exist two diffeomorphisms $\Theta_1:F^{-1}(p)\rightarrow F^{-1}(q)$ and $\Theta_2:F^{-1}(q)\rightarrow F^{-1}(z)$. $\Theta_2\circ\Theta_1$ does the job.
Let's suppose $N$ is connected, so there's a path from $p$ to $z$. We can cover this path with a finite set of open balls that interset one after the other. So the final result is proven.
