= Solution
Fix $q_1$ and choose $U$ as in part c. For $q_2\in U$, choose the compactly supported $w$ moving $q_1$ to $q_2$, and lift it by part b to $v$. The support of $v$ is contained in
$$
F^{-1}(\operatorname{supp}w),
$$
which is compact because $F$ is a <proper map>. Hence $v$ is compactly supported and complete. If $\Phi^t$ and $\Psi^t$ are the flows of $v$ and $w$, then
$$
\frac d{dt}F(\Phi^t(p))
=D_{\Phi^t(p)}F(v)=w(F(\Phi^t(p))).
$$
Uniqueness of integral curves gives
$$
F\circ\Phi^t=\Psi^t\circ F.
$$
Therefore the diffeomorphism $\Phi^1$ maps $F^{-1}(q_1)$ onto $F^{-1}(q_2)$. Every equivalence class is open. Its complement, being a union of the other open classes, is also open; thus each class is clopen. If $Y$ is connected, there is only one class. This proves the fiber-diffeomorphism conclusion of the <Ehresmann fibration theorem>.
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