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We use discrete Morse theory to give a new proof of the Degree Theorem in Auter space A_n. There is a filtration of A_n into subspaces A_n,k using the degree of a graph, and the Degree Theorem says that each A_n,k is (k-1)-connected. This result is useful, for example to calculate stability bounds for the homology of Aut(F_n). The standard proof of the Degree Theorem is global in nature. Here we give a proof that only uses local considerations, and lends itself more readily to generalization.arXiv:0907.4642v4 fatcat:apuvfviuevfbxb4thf4ojuajpe