For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$u \in \mathcal {L}^{\prime }$\end{document}u∈ℒ′ terms, the sums are exactly those we analyzed for double anchored distances; thus, the additivity is already proved. Since the sum of two additive distances is additive, it suffices to prove additivity for u∈{a,b,c,d} cases, marked as L,R1, and R2 above.