The proof of the uniqueness of the conformal parallel slit transformation is given in and . Applying the inverse of the Joukowsky transform to the horizontal slit domain, it can be assumed that is a domain bounded by the unit circle and contains analytic arcs and isolated points (the images of other the inverse of the Joukowsky transform under the other parallel horizontal slits). Thus, taking a fixed in , there is a univalent mapping = = with image a horizontal slit domain. Suppose that is another uniformizer with = . The images under or of each have a fixed "y"-coordinate so are horizontal segments. On the other hand = is holomorphic in . If it is constant, then it must be identically zero since = 0. Suppose is non-constant. Then by assumption are all horizontal lines. If is not in one of these lines, Cauchy's argument principle shows that the number of solutions of = in is zero (any will eventually be encircled by contours in close to the 's). This contradicts the fact that the non-constant holomorphic function is an open mapping.