For a trivial fiber G-bundle transition functions $t_{UV}: U \cap V \to G$ can be defined as $t_{UV}(x) = f_U(x)f_V^{-1}(x)$, for continuous $f_U,f_V$

Question

The relevant definitions can be found on the wiki page. I hoped that I could prove the statement by showing that a maximal G-Atlas for a trivial map must contain a chart $(B, \varphi_i)$. It then would follow that functions $f_U$ can be defined as transition maps to $B$; that is, $f_U := t_{UB}$, and the result would follow from the properties of transition maps. However, I don't know how to prove that any maximal G-Atlas of a trivial map must contain a $(B, \varphi_i)$ chart, and I even think that it might be wrong. Thus, I'd like to prove that such a chart is necessary in every maximal G-Atlas or to get a hint how to prove the statement in the title using other ways.