Let $B$ be a smooth manifold with open cover $(U_i)_{i\in I}$, let $F$ be a smooth manifold, and let $G$ be a Lie group acting smoothly on $F$. Suppose smooth maps $g_{ij}:U_i\cap U_j\to G$ are given and satisfy $g_{ii}(b)=e$ and $g_{ij}(b)g_{jk}(b)=g_{ik}(b)$ for all $b\in U_i\cap U_j\cap U_k$. Let $E$ be the quotient of $\bigsqcup_i U_i\times F$ by $(i,b,y)\sim (j,b,g_{ji}(b)y)$, and give $E$ the atlas topology induced by the quotient local charts $U_i\times F\to E$. Assume this atlas topology is Hausdorff and second-countable. Then $E$ has a natural smooth fibre bundle structure over $B$ with fibre $F$ and structure group $G$.