it from "interior variation"), he derived a differential equation and inequality, that relied on a measure-theoretic characterisation of straight-line segments due to Ughtred Shuttleworth Haslam-Jones from 1936. Haslam-Jones' proof was regarded as difficult and was only given a satisfactory proof in the mid-1970s by Schober and Campbell–Lamoureux. gave a proof of uniformization for parallel slit domains which was similar to the Riemann mapping theorem. To simplify notation, horizontal slits will be taken. Firstly, by Bieberbach's inequality, any univalent function = with in the open unit disk must satisfy || ≤ 2. As a consequence, if = is univalent in | | > , then | | ≤ 2 | |: take , set = for in the unit disk, choosing so the denominator is nowhere-vanishing, and apply the Schwarz lemma. Next the function = is characterized by an "extremal condition" as the unique univalent function in of the form that maximises : this is an immediate consequence of Grönwall's area theorem, applied to the family of univalent functions in . 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