Now consider the “single anchored” distance matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D^{*}_{v}$\end{document}Dv∗ on the leaf-set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {L}-\{v\}$\end{document}ℒ−{v} (for a single \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$v\in \mathcal {L}$\end{document}v∈ℒ). To prove additivity of the single anchored distance matrix, we need to prove the following four point condition: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{array}{@{}rcl@{}} & \sum_{u \notin \{a,b\}} D^{\prime}_{uv}[a,b] + \sum_{u \notin \{c,d\}} D^{\prime}_{uv}[c,d] &< \\ & \sum_{u \notin \{a,d\}} D^{\prime}_{uv}[a,d] + \sum_{u \notin \{a,b\}} D^{\prime}_{uv}[b,c] &= \\ & \sum_{u \notin \{a,c\}} D^{\prime}_{uv}[a,c] + \sum_{u \notin \{b,d\}} D^{\prime}_{uv}[b,d] \end{array} $$ \end{document}∑u∉{a,b}Duv′[a,b]+∑u∉{c,d}Duv′[c,d]<∑u∉{a,d}Duv′[a,d]+∑u∉{a,b}Duv′[b,c]=∑u∉{a,c}Duv′[a,c]+∑u∉{b,d}Duv′[b,d]