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\layout Proof Therefore, each equivalence class of the adjacency relation corresponds to an interval in \begin_inset Formula $\{1,\ldots,k\}$ \end_inset . More precisely, let \begin_inset Formula $1=\bindex{1}<\bindex{2}<\cdots<\bindex{x}<\bindex{x+1}=k+1$ \end_inset be indices \the_end