Let $G$ be a Lie group, let $P\to M$ be a smooth principal $G$-bundle with a smooth connection form $\omega$, and let $U\subset M$ be a connected coordinate neighbourhood over which $P$ is trivialised by a smooth local section $s:U\to P$. Let $A:=s^*\omega\in\Omega^1(U;\mathfrak g)$ be the corresponding local connection form, and let $F_A$ be its local curvature form. If $F_A$ vanishes on $U$, then after possibly shrinking $U$ to a simply connected coordinate neighbourhood, there is a smooth local section for which the connection form is zero.