Many modern proofs of the (ineffective) finiteness of solutions of the $S$-unit equation $x+y=1$ use Roth's theorem. In particular it is used Lang's version of Roth's theorem which takes in account multiple places. There are also other proofs using Beukers methods.
It seems that the original proof (for a number field) is due to Mahler, in the paper "Zur Approximation algebraischer Zahlen. I. (Über den größten Primteiler binärer Formen), 1933. He generalises what Siegel had already done over $\mathbb Q$.
I cannot read German and moreover the paper seems long and full of computations. My question here is very simple:
In 1933 Mahler couldn't use Roth's theorem, because it didn't exist. So, from which Diophantine approximation result does he deduce the theorem on the $S$-unit equation?
Often in literature it is said that Mahler proved the theorem implicitly, and that the first explicit proof was given by Lang in 1960. I suppose that they refer to the proof he gives in his book "Fundamentals of Diophantine Geometry".
