Lebesgue's Theorem...

I've had no success puzzling over Riesz/Nagy proof of the existence of a finite derivative for the class of monotone functions. Alas, searches on the internet keep leading to the standard proof involving Vitali's covering theorem. I imagine Vitali's work is embedded in the Riesz's version, but the Riesz/Nagy proof is so simple (except for the inequality I mentioned) that the unexplained equality is getting quite frustrating!