 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 inwhich 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 $w$, thenUniqueness of integral curves givesTherefore 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.