Let $M$ be a smooth $m$-manifold, let $N$ be a smooth $n$-manifold, and let $F:M\to N$ be a smooth map. Suppose $\operatorname{rank}_qF=r$ for every $q$ in a neighbourhood of $p\in M$. Then there are charts $(U,\varphi)$ around $p$ and $(V,\psi)$ around $F(p)$, with $\varphi(p)=0\in\mathbb{R}^m$ and $\psi(F(p))=0\in\mathbb{R}^n$, such that $\psi\circ F\circ\varphi^{-1}$ has the local form \begin{align*} (x_1,\ldots,x_m)\mapsto(x_1,\ldots,x_r,0,\ldots,0). \end{align*}