Untangling two simple closed curves on a surface

Question

Let $S$ be a smooth surface and $\gamma_1, \gamma_2$ be two transversal simple closed curves on it. Suppose moreover that there exists a simple closed curve $\gamma_1'$ on $S$ isotopic to $\gamma_1$ and such that $\#(\gamma_1\cap \gamma_2)>\#(\gamma_1'\cap \gamma_2)$.

Question. Is it true that there is a disk on $S\setminus (\gamma_1\cup\gamma_2)$ whose boundary is composed of one arc of $\gamma_1$ and one arc of $\gamma_2$?

Note that in case such a disk exists, one can construct an isotopy of $\gamma_1$ that would decrease the number of intersections of $\gamma_1$ with $\gamma_2$ by two.

Answer 1

This is also proved as Lemma 3.1 in Joel Hass and Peter Scott, Intersections of curves on surfaces, Israel Journal of Mathematics 51 (1985), 90–120. https://doi.org/10.1007/BF02772960