We prove that Eq. (4) returns the sum of internal branch lengths on the path from a to b on the tree T (we denote this by D T ab). The theorem immediately follows because a distance matrix compatible with the tree T has to be by definition additive and compatible with it (note that the theorem also claims that D ′′ gives internal branch lengths). For simplicity, we prove with α=1; extension to other values is simple. If a and b are not sisters in T, there exists an anchor pair (u,v) with quartet topology a u.b v and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tau (a,b,u,v)=D^{T}_{ab}$\end{document}τ(a,b,u,v)=DabT; to find such u and v, the following procedure can be followed. Pick u arbitrarily from the sister group of a after rooting T on b and pick v arbitrarily from the sister group of b after rooting T on a. With this choice, it’s easy to see that τ(a,b,u,v) becomes simply the sum of internal branches between a and b; thus, from the first case of Eq. (1), we have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D^{\prime }_{uv}[a,b]-\beta =\frac {\tau (a,b,u,v)}{\alpha }+\beta -\beta =D^{T}_{ab}$\end{document}Duv′[a,b]−β=τ(a,b,u,v)α+β−β=DabT. Moreover, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D^{\prime }_{wz}[a,b]-\beta $\end{document}Dwz′[a,b]−β for two other anchors w,z cannot be bigger than \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D^{T}_{ab}$\end{document}DabT. That is because if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ab.wz\in \mathcal {Q}^{T}$\end{document}ab.wz∈QT, then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D^{\prime }_{wz}[a,b]<\beta $\end{document}Dwz′[a,b]<β; else, τ(a,b,w,z) will give the length for a subpath from a to b. Thus, the max function in Eq. (4) returns \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D^{T}_{ab}$\end{document}DabT, as desired. When (a,b) are sisters, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D^{T}_{ab}=0$\end{document}DabT=0; also \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D^{\prime }_{uv}[a,b]<0$\end{document}Duv′[a,b]<0 for any (u,v), and thus, the max function returns D ′′[a,b]=0; this is what we want, since for sisters, the length of the internal branch length is zero. Thus, as desired, Eq. (4) always returns the length of the internal branches in the T between a and b; this completes the proof. □