Theorem ā perfect matchings M and Mā², ā transformation of length at most 2ālg(n)ā between M and Mā² Proof. S the set of 2n points. By lemma iii, ā perfect matchings M and Mā² : M = M0, M1, M2,..., Mk = N(S) and Mā² = Mā² 0, Mā² 1, Mā² 2,..., Mā² kā² = N(S) with k , kā² ā¤ ālg(n)ā. Thus M0, M1, M2,..., Mk = Mā² kā² ,..., Mā² 2, Mā² 1, Mā² 0 = Mā² is a transformation of length at most 2ālg(n)ā. Disjoint Compatible Perfect Matchings 12 / 16