I am trying to understand the proof of the following corollary in Hirsch's Differential Topology :
Two $C^r$ vector bundles $\eta_0$ and $\eta_1$ over a paracompact base space $B$ are $C^r$ isomorphic if and only if there is a $C^r$ vector bundle $\eta$ such that $\eta_i \cong (\eta|_{B\times i })$, $i=0,1$.
Now I am having touble with $\Leftarrow$. By the covering homotopy theorem we know that $\eta \cong (\eta|_{B\times 0})\times I$, but I don't know what to do with this, I am a bit confused with the isomorphisms that are going on.
Any help with this is aprecciated. Thanks in advance.