This consequence is striking because it enables one to define the various conformal structures carried by X using functions defined only on all of X and properties only of a topological or algebraic sort. Differentiability is never mentioned. Since the algebra in question contains a member, h, which is light and interior, one concludes by Stoilow's theorem that h is locally topologically equivalent with power functions, i.e., h is a ramified cover of the sphere. Thus there exists a unique conformal structure for X relative to which h is meromorphic. For proof, see, e.g., [2, pp. 119-120]. If, relative to this structure, c is an analytic homeomorphism sending a disk into X, then the algebra generated by h(c), f(c), and g(c) consists entirely of interior mappings (i.e., of mappings which send open sets onto open sets) and has one member, h(c), which is known to be meromorphic. If these functions have no poles, then by Rudin's theorem (see Theorem 2 below) they are all analytic. Consequently, f and g are analytic (except for poles) on X relative to the same conformal structure. Riemann's theorem guarantees then that they are meromorphic on X relative to that structure. This completes the proof, for the converse is standard.