I am looking for a reference for the following admittedly imprecise statement:

Any projective invariant of n points in the projective plane may be expressed as a function of well-chosen cross-ratios.

By *projective invariant* I mean a rational function defined on the set of $n$-tuple of distinct points in the projective plane on an arbitrary field $K$, invariant under the action of the projective group of transformations.
This is a folklore result that is often stated without proof (e.g. in Efimov, *higher geometry*) but I can't find a reference providing a precise statement and a proof.

on a line. Here you have $n$ general points in the projective plane, so I expect that non even three of them are collinear. What cross-ratios are you talking about? $\endgroup$