In Chapter 10 about Teichmüller spaces of Farb and Margalit's "A Primer to Mapping Class Groups", the length function $$\ell: \operatorname{Teich}(S) \to \mathbb{R}^\mathcal{S}$$ is described, where $S$ is a compact surface with finitely many punctures and $\chi(S) < 0$ and $\mathcal{S}$ denotes the set of isotopy classes of essential simple closed curves in $S$.

It is already impressive that the map $\ell$ is injective (which they also remark), however they prove the much stronger $9g-9$ theorem.

**Question:** Is there a (simpler) proof that $\ell$ is injective **without** showing that the length function is actually determined by its value on only $9g-9$ simple closed curves?

I've searched for a bit now, and came across e.g. these lecture notes, where Theorem 3.3 claims that the map is injective. The reference given is "Travaux de Thurston sur les surfaces", in which Theorem 7.9 states that $\ell$ is injective. However, it is not proven independently from the $9g-9$ theorem after all.

isproven, because it is a corollary of the $9g-9$ theorem. $\endgroup$