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A principal G-bundle , $\pi: P \to B$ is locally equivalent to a product. If Depending on who you choose ask, this is either part of the definition, or a suitable short lemma. It means that there is a cover of B by open sets U, and choose trivializations of $\pi$ on these open sets (i.e., together with bundle isomorphisms $\alpha_U: \pi^{-1}U \to G \times U$)U$that are both G-equivariant, then and induce transition sections in G: see Wikipedia. The associated bundle construction is a quotient of$\alpha_U P \times_\rho id$is times \mathbb{C}^n$ by an isomorphism from equivalence relation given by the restriction actions of G on the associated bundle over U to left and right factors. Since $\alpha_U$ is G-equivariant, the trivial map $\alpha_u \times id: \pi^{-1}U \times \mathbb{C}^n \to G \times U \times \mathbb{C}^n$ is also a G-equivariant bundle .map, so you get bundle isomorphisms on the quotient bundles: $\alpha_u \times_\rho id: \pi^{-1}U \times_\rho \mathbb{C}^n \to U \times \mathbb{C}^n$.
By the definition of principal G-bundle, $\pi: P \to B$ is locally equivalent to a product. If you choose a suitable cover of B by open sets U, and choose trivializations of $\pi$ on these open sets (i.e., isomorphisms $\alpha_U: \pi^{-1}U \to G \times U$), then $\alpha_U \times_\rho id$ is an isomorphism from the restriction of the associated bundle over U to the trivial bundle.