Let $\mathcal{F}$ be a rank-$r$ locally-free sheaf on a ringed space $(X,\mathcal{O}_X)$ with a cover $\{U_\alpha\}_\alpha$ such that $\mathcal{F}|_{U_\alpha}\cong(\mathcal{O}_X|_{U_\alpha})^r$.
Is it true that $\mathcal{F}|_{U_{\alpha_0\cdots\alpha_p}}\cong(\mathcal{O}_X|_{U_{\alpha_0\cdots\alpha_p}})^r$ for all $U_{\alpha_0\cdots\alpha_p}=\bigcap_{i=0}^p U_{\alpha_i}$?
This is true, it follows from the fact you can compose restrictions. Indeed, if $\mathcal{F}_{|U}\simeq(\mathcal{O}_X)_{|U}^r$ then $$\mathcal{F}_{|V}=(\mathcal{F}_{|U})_{|V}\simeq(\mathcal{O}_X)_{|V}^r$$