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Let S be a multiplicative topological semigroup with identity e. Suppose D is an open subset containing e and A is a homeomorphism from D onto a Banach space B with h(e) = 0. Define the function P by P(x, y) = h[h~l(x) ■ h~l(y)]. A new implicit function theorem is applied to the function P to show the existence of inverses and square roots of elements in a neighborhood of the identity. It is assumed that P satisfies the following condition: There exist a one-one function A from a subset of Bdoi:10.1090/s0002-9939-1976-0399338-2 fatcat:2ds5icqp7zhdbl54coz4zgg7te