Let $C_1, \dots, C_n$ be a family of disjoint simple curves in a surface $\Sigma$. If $C$ is any simple curve in $\Sigma$, it turns out that we can map $C$ to a curve $C'$ (via a homeomorphism of $\Sigma$) such that $C'$ only intersects each $C_i$ at most twice. I believe I have a proof of this result that works for all surfaces, but I'm pretty sure this is classical stuff.

Indeed, in John Stillwell's book *Classical Topology and Combinatorial Group Theory* he mentions that the above result was proven by Lickorish (1962) for orientable surfaces. In the case of orientable surfaces, the homeomorphism can in fact be achieved via Dehn twists and isotopies. Unfortunately, the Lickorish proof doesn't work for non-orientable surfaces.

**Question:** Can someone please provide a reference of the above result for non-orientable surfaces?