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admissible function analytio arc which separates bdyG bdyH bdyK Cauchy-Riemann equations circular covering circular region containing class C^1 component conformal cover conformal mapping continuous boundary values Definition denotes the set Dirichlet integral Dirichlet principle dx dy end-point exists a neighborhood exists a positive exists a region finite number function almost class Green's formula harmonio Hausdorff space Hence homeomorphism interior intersects G Lemma Let f Let G limit point line integral manifold minimizing sequence olass olosed oover open arc ordered triple orientation pairwise disjoint perfectly separable Riemann plane points of f pole of order positive integer preferred region Proof provided f provided there exists quasi-simple real numbers real-valued function defined real-valued funotion region G separable Riemann surface separates GQ separation axiom simple closed curve Sinoe slit region smoothed modification spaoe Stoilow Theorem topological triangulable triple h uniformizer g unit circle unit interval