I'm having some trouble with one part of Bott–Tu Theorem 6.8, which proves that homotopic maps induce isomorphic vector bundles under pullback.
The setup is that $\pi:Y\times I\to Y$ is the projection, $f:Y\times I\to X$ is a homotopy between $f_0$ and $f_1$, and $E$ is a vector bundle over $X$. Let $f_{t_0}^{-1}E$ be isomorphic to the vector bundle $F$.
Now there are bundles $f^{-1}E$ and $\pi^{-1}F$ over $Y\times I$. We also have a section of $\text{Hom}(f^{-1}E,\pi^{-1}F)$ over $Y\times t_0$, where the bundles are isomorphic. Covering $Y\times\{t_0\}$ with finitely many trivializing open sets for the Hom-bundle (we assume $Y$ is compact), the authors then say the following: "As the fibers of $\text{Hom}(f^{-1}E,\pi^{-1}F)$ are Euclidean spaces, the section over $Y\times t_0$ may be extended to a section of $\text{Hom}(f^{-1}E,\pi^{-1}F)$ over the union of these open sets."
I don't understand why this is true, and why this is dependent on the fibers being Euclidean (as opposed to $\text{Hom}(f^{-1}E,\pi^{-1}F)$ being a general fiber bundle). I feel that I am missing something really obvious here.