John M lee, Introduction to smooth manifolds, Exercise 5-4:
Let $M$ be a smooth manifold and let $\{U_α\}_{α∈A}$ be an open cover of $M$. Suppose for each $α, β ∈ A$ we are given a smooth map $τ_{αβ} : U_α∩U_β → GL(k,\mathbb{R})$ such that $τ_{αβ}(p)τ_{βγ}(p) = τ_{αγ}(p)$, $p∈ U_α ∩ U_β ∩ U_γ$ is satisfied for all $α, β, γ ∈ A$. Show that there is a smooth rank $k$ vector bundle $E → M$ with smooth local trivializations $Φ_α : π^{−1}(U_α) → U_α×\mathbb{R}^k$ whose transition functions are the given maps $τ_{αβ}$. [Hint: Define an appropriate equivalence relation on $\coprod_{α∈A}(U_α × \mathbb{R}^k)$, and use the bundle construction lemma.]
But the bundle construction lemma has a different hypothesis. It uses $E= \coprod _{p \in M} E_p$ where $E_p$ is a real vector space. How can I use the lemma?