On page 4 of this Master's thesis, we have:
Lemma 2.2. Let $F$ be a compact surface, let $D_1, \ldots , D_k$ be a collection of pairwise-disjoint disks embedded in $F$ and let $D_1', \ldots , D_k'$ be a second collection of disjoint embedded disks. There is a homeomorpism $\varphi : F \rightarrow F$, isotopic to the identity, such that $\varphi$ sends each disk $D_i$ onto the disk $D_i'.$
The proof is just a reference to C. P. Rourke and B. J. Sanderson's 1972 edition of Introduction to piecewise-linear topology. I'm looking at the Springer Study edition of this same book, but the one I'm looking at was published in 1982. In any event, there's no hint of this theorem being in there. There's something called the "Disc Theorem" on page 44 that seems vaguely related, but only works for discs in the interior of the manifold and which therefore isn't directly applicable.
Question. What's going on here? Is this lemma legitimate, and if so, how do we know it's true?