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. 
