For terms of type (III) and (II), the additivity is already proved in double and single anchored cases, respectively. Thus, we need to prove additivity only for terms of type (I), which have no anchors. Let x=τ(a,b,c,d). \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{array}{*{20}l} &2D^{\prime}_{ab}[c,d] = 2[\beta-f(x)]< 2\beta<2[\beta+\alpha x] = \\ & 2D^{\prime}_{ad}[b,c] = 2D^{\prime}_{ac}[b,d] \end{array} $$ \end{document}2Dab′[c,d]=2[β−f(x)]<2β<2[β+αx]=2Dad′[b,c]=2Dac′[b,d]