everyone.
I am studying D.J Saunders's book The Geometry of Jet Bundles. On proposition 1.1.14, a proof is given that the structure of the total space $E$ of a bundle $(E,\pi,M)$ depends on those of its base space $M$ and typical fiber $F$. The statement of the proposition is as follows:
"Let $M$ and $F$ be manifolds, $E$ a set, and $\pi: E \rightarrow M$ a function such that, for each $p \in M$, $\pi^{-1}(p)$ has the structure of a $n$-dimensional manifold. Suppose also that, for each $p \in M$, there is a neighborhood $W_p$ of $p$ and a bijection $t_p: \pi^{-1}(W_p) \rightarrow W_p \times F$ satisfying
$pr_1 \circ t_p =\pi\restriction_{\pi^{-1}(W_p)}$
for each $q \in W_p$, $pr_2 \circ t_p\restriction_{\pi^{-1}(q)}: \pi^{-1}(q) \rightarrow F$ is a diffeomorphism.
Then $E$ may be given a unique structure as a manifold such that $\pi$ becomes a bundle and the maps $t_p$ become local trivialisations".
Now my problem is the following. Saunders claims that it is sufficient to prove that
$t_q \circ t_p^{-1}: (W_p \ \cap \ W_q \times F \rightarrow W_p \ \cap \ W_q \times F)$
is smooth in order to prove that $y_p=(x,u) \circ t_p$ is a coordinate system around $a \in \pi^{-1}(p)$. Here $x : W \rightarrow \mathbb{R}^{m}$ is a coordinate system around $p$ and $u: V \rightarrow \mathbb{R}^{n}$ is a coordinate system around $pr_2(t_p(a)) \in F$.
So far so good, I understand. After that, he says that to prove the above, we first note that for each $r \in W_p \ \cap \ W_q $, the map $t_q \circ t_p^{-1}\restriction_{\{r\} \times F}$ induces a diffeomorphism of $F$ in itself. I understand this also. But then he says that a consequence from this is that the following map is smooth:
$W_p \ \cap \ W_q \times F \rightarrow F$
$(r,c) \mapsto pr_2(t_q \circ t_p^{-1}\restriction_{\{r\} \times F}(c))$.
Now, I don't understand why this map is smooth in $r$. Could someone please help?