I'm studying the $H$-structures on principal bundles, here is the definition:
Let $p:E\rightarrow B$ be a principal $G$-bundle and let $H<G$ a closed subgroup. Then $p:E\rightarrow B$ is said to have a reduction of the structure group to H from G if it is isomorphic to a locally trivial fibration with transition functions that takes values in H.
In particular this means that if it's the case, there exists a principal H-bundle E' and an isomorphism $$P\equiv E'\times_{H}G\stackrel{\sim}{\rightarrow} E$$ My question is:-How do I know that P has transition functions in H?