(Lifting lemma): Let M be a Hausdorff space and let be the projection map. Let be a continuous map. Then there is a continuous map such that .
Proof of Lemma 5.1. We use induction on n. The case n=1 is trivial, so assume n > 1. By Sublemma 5.1, it suffices to find a lift on each connected component I of . By Sublemma 5.5 it suffices to show that any has a neighborhood on which there is a lift.Suppose () are the distinct elements of the multiset , occurring with multiplicities respectively. Since M is Hausdorff, there exist pairwise disjoint neighborhoods of . Let N be a closed interval neighborhood of such that implies . Then on N, we can lift f to a path in since the projection
restricts to a homeomorphism on the projections of . By the inductive hypothesis applied to each of the k coordinates of , we can lift to a path in as desired.