Reading functions in local coordinates (vector bundles) I'm currently studying bundles from Husemaller and don't understand the proof of Proposition $1.5$ p.$25$.
The argument is to show the continuity property holds on an open covering, i.e the one composed by $p^{-1}(U)$. What I don't get is how to relate the definition of vector bundles in a way that I could use the $h$-isomorphism of the definition and the maps in the proposition which are $a,s$.
My attempt for $a$ since $s$ should be similar: I thought this is the interpretation of reading a map in charts so I end up having the following diagram
$\require{AMScd}$
\begin{CD}
    U \times F^k \times F^k @>{h}>> p^{-1}(U) \oplus p^{-1}(U) = q^{-1}(U)\\
    @V{\tilde{a}}VV @V{a}VV\\
    U \times F^k @>{h}>> p^{-1}(U)
\end{CD}
And I thought that it would make sense if $\tilde{a}(u,x,y) = (u,x+y)$ which is continuos if $F = \mathbb{R},\mathbb{C}, \mathbb{H}$ which are the cases I'm working with.
The problem here is that I don't know how to create the diagram and whether the maps are explicit in order to have a commutative diagram.
Any help in order to clarify this detail and understand this concept would be appreciated.

Let me write $E|_U$ for $p^{-1}(U)$. Let $h:E|_U \to U \times F^k$ be a trivialisation of $E|_U$, that is $h$ is a homeomorphism and $h_p : E_p \to \{p\}\times F^k$ is a vector space isomorphism. Now we can build a trivialsiation  of $(E\oplus E)|_U$:
$$\tilde h: (E\oplus E)|_U \to U \times F^k \times F^k \\  (v\oplus w) \mapsto \left(pr_U(h(v)), pr_{F^k}(h(v)), pr_{F^k}(h(w))\right)$$
Let $$\tilde a: U \times F^k \times F^k \to U \times F^k \\ (u,x,y) \mapsto (u,x+y)$$ and $$a: (E\oplus E)|_U \to E|_U\\ (v\oplus w) \mapsto (v + w)$$ Now check that $h\circ a = \tilde a \circ \tilde h$. Then $a$ is continous if and only if $\tilde a$ is continous.
