We now prove the additivity for the all-pairs matrix. Equation (3) has three types of terms: {u,v}∩{a,b,c,d} may have (I) both anchors, (II) one anchor, or (III) none. The four point condition can be written: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $${{}\begin{aligned} \overbrace{2 D^{\prime}_{ab}[c,d]}^{I} + &\overbrace{\sum\limits_{v \in \mathcal{L}^{\prime}} \sum\limits_{u \in \{c,d\}} D^{\prime}_{uv}[a,b] + \sum\limits_{u \in \{a,b\}}D^{\prime}_{uv}[c,d]}^{II} + & \\ &\overbrace{\sum\limits_{u,v \in \mathcal{L}^{\prime}} D^{\prime}_{uv}[a,b] + D^{\prime}_{uv}[c,d]}^{III} &< \\ 2 D^{\prime}_{ad}[b,c] +& \sum\limits_{v \in \mathcal{L}^{\prime}} \sum\limits_{u \in \{b,c\}} D^{\prime}_{uv}[a,d] +\sum\limits_{u \in \{a,d\}} D^{\prime}_{uv}[b,c] +&\\ &\sum\limits_{u,v \in \mathcal{L}^{\prime}} D^{\prime}_{uv}[a,d] + D^{\prime}_{uv}[b,c] &=\\ 2 D^{\prime}_{ac}[b,d] + &\sum\limits_{v \in \mathcal{L}^{\prime}} \sum\limits_{u \in \{b,d\}} D^{\prime}_{uv}[a,c] +\sum\limits_{u \in \{a,c\}} D^{\prime}_{uv}[b,d] +&\\ &\sum\limits_{u,v \in \mathcal{L}^{\prime}} D^{\prime}_{uv}[a,c] + D^{\prime}_{uv}[b,d] \end{aligned}} $$ \end{document}2Dab′[c,d]⏞I+∑v∈ℒ′∑u∈{c,d}Duv′[a,b]+∑u∈{a,b}Duv′[c,d]⏞II+∑u,v∈ℒ′Duv′[a,b]+Duv′[c,d]⏞III<2Dad′[b,c]+∑v∈ℒ′∑u∈{b,c}Duv′[a,d]+∑u∈{a,d}Duv′[b,c]+∑u,v∈ℒ′Duv′[a,d]+Duv′[b,c]=2Dac′[b,d]+∑v∈ℒ′∑u∈{b,d}Duv′[a,c]+∑u∈{a,c}Duv′[b,d]+∑u,v∈ℒ′Duv′[a,c]+Duv′[b,d]