Without the refinement involving m, non-bijective proofs of this theorem have been given by Carlos Nicolas ,

where By definition, t j (P) = t j (Q) and m j (P) = m j (Q) for j / ? {i ? 1, i}. By Lemma 2.11, the number of east steps where P i and P i+1 (respectively P i?1 ) coincide equals the number of east steps where ?(P i ) and P i?1 (respectively P i+1 ) coincide, so t i (P) = t i?1 (Q) and t i?1 (P) = t i (Q). It remains to show that m i?1 (P) + m i (P) = m i?1 (Q) + m i (Q), which we do in ,

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