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\begin{proof}
$\Gamma = (V, E), \mathcal{D}(\Gamma)$ its neighbourhood design.\\
$[P, Q] \in E$ is a point of the line graph $L(\Gamma)$ and $\overline{[P, Q]}$ is a block of $ \mathcal{D}(\Gamma)$:
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$\overline{[P, Q]} = \lbrace [P, R] \ \vert \ R \neq Q \rbrace \ \bigcup \ \lbrace [R, Q] \ \vert \ R \neq P \rbrace$
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thus $v^{\overline{[P,Q]}} +v^{\overline{[R,S]}} - v^{\overline{[P, S]}} - v^{\overline{[Q, R]}} = -2(v^{[P,Q]} + v^{[R,S]} - v^{[P,S]} - v^{[Q, R]})$.\blacksquare
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