A direct proof of Stahl's theorem for a generic class of algebraic functions

Under the assumption of the existence of Stahl's $S$-compact set, we give a short proof of the existence of limit zero distribution of Padé polynomials and convergence in capacity of diagonal Padé approximants for a generic class of algebraic functions. The proof is direct but not from the opposite as Stahl's original proof is. The generic class means in particular that all the branch points of the multi-sheeted Riemann surface of the given algebraic function are of the first order (i.e., we assume the surface is such that all the branch points are of square root type).
We do not use the orthogonality relations at all. The proof is based on the maximum principle only.
It is supposed to discuss in short in what way the proposed approach can be extended to Hermite–Padé polynomials case.