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The article is about the analysis of semi-local convergence of a seventh-order iterative method used for finding the roots of a nonlinear equation in Banach spaces. In this article, the imposed hypotheses is amiable than the well-known Lipschitz and Hölder continuity conditions. The R-order convergence of the considered scheme is proved to be at least 4+3q. An approximate apriori error bound for this method is also elaborated and the domain of existence and uniqueness of the solution in the convergence theorem. Two numerical illustrations have been worked out to exhibit the virtue of the developed theory.