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Let K be the scalar field of the first orthomodular (or Form Hilbert) space,described by H. Keller in 1980. It has a non-Archimedean order, an infiniterank valuation compatible with the order as well as an explicitly definedultrametric, all of which induce the same topology. We study analytic functionsdefined on valued field K, and we will establish an invertibility local theoremfor these functions as an application of Banach fixed point theorem.

On analytic functions in an ordered field with an infinite rank valuation