Continuity of translation function in fibers

Question

Let $P$ be a principial $G$-bundle (over base space $B$). Then $G$ acts freely and transitively on the fibers of $P$. Therefore if we take $p,p' \in P_x$ (the fiber over $x$) we can find unique $g \in G$ such that $p'=p \cdot g$. Let us call this element $\tau(p,p')$. This gives rise to the function $\bigcup_{x \in B} (P_x \times P_x) \to G$. How to prove that this function is continuous? For my interests it is enough to know this for locally compact groups $G$.

Answer 1

By trivializing $P$ and fixing a point in a fiber your map becomes simply: \begin{equation} \begin{aligned} G\times G&\to G\\ (g, h)&\mapsto g^{-1}h \end{aligned} \end{equation}

which is clearly continuous.

Notes: 1) no assumption on G is needed. 2) In some treatments (e.g. Husemoller's Fibre Bundles) the continuity of the translation function is part of the definition.