Let $M$ be a smooth finite-dimensional second-countable Hausdorff manifold, let $N\subset M$ be a closed embedded submanifold, and let $i:N\to M$ be the inclusion map. Let $k\ge 1$, let $V\subset M$ be an open neighbourhood of $N$, and let $\alpha\in\Omega^k(V)$ satisfy $d\alpha=0$ and $i_V^*\alpha=0$, where $i_V:N\to V$ is the inclusion map. Then there exist an open neighbourhood $U\subset V$ of $N$ and a form $\beta\in\Omega^{k-1}(U)$ such that

\begin{align*} d\beta=\alpha|_U. \end{align*}

Moreover, $\beta$ may be chosen so that

\begin{align*} i_U^*\beta=0, \end{align*}

where $i_U:N\to U$ is the inclusion map.