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2. Proof of Theorem 1. Assume that h(z) has an essential singularity at a point 2* of ER. Then in an arbitrary small neighborhood of the point 2* h(z) takes every value infinitely often excepting at most two. Hence all the points of Es excepting at most two are taken by h(z). This contradicts the into-ness of h(z). Thus there is no essential singularity of h at ER. The same is true for z=oo. Hence h(z) must be a rational function of z in the ^-sphere. Next we prove that h(z) is a polynomial of z. Assume that h(z) has a pole at some point in the finite 2-plane. Then a fixed neighborhood of this point is mapped around Z=&> by h(z) and its image by h(z) contains a neighborhood of Z=oo, which contains at least a point of Es. This contradicts the into-ness of h(z) in DR. This implies that h(z) is a polynomial.

On rigid analytic mappings among surfaces $\{e^{w}=f(z)\}$