on all overlaps where the expressions are defined. Then there exists a smooth principal right $G$-bundle $\rho:P\to M$ with local sections $s_i:U_i\to P$ satisfying $s_j(x)=s_i(x)g_{ij}(x)$ for every $x \in U_i \cap U_j$. The bundle is unique up to principal bundle isomorphism inducing the identity on $M$.