To prove now that the multiply connected domain ⊂ ℂ ∪ {∞} can be uniformized by a horizontal parallel slit conformal mapping = , take large enough that lies in the open disk || . For , univalency and the estimate | | || imply that, if lies in with | | , then | | . Since the family of univalent are locally bounded in \ {∞}, by Montel's theorem they form a normal family. Furthermore if is in the family and tends to uniformly on compacta, then is also in the family and each coefficient of the Laurent expansion at ∞ of the tends to the corresponding coefficient of . This applies in particular to the coefficient: so by compactness there is a univalent which maximizes . To check that = is the required parallel slit transformation, suppose "reductio ad absurdum" that = has a compact and connected component of its boundary which is not a horizontal slit. Then the complement of in ℂ ∪ {∞} is simply connected with ⊃ . By the Riemann mapping theorem there is a conformal mapping = such that is ℂ with a horizontal slit removed. So = and hence by the extremality of . Thus . On the other hand by the Riemann mapping theorem there is a conformal mapping = from || onto . Then = . By the strict maximality for the slit mapping in the previous paragraph , so that > 0. The two inequalities for are contradictory.