The proof[1] is based on the notion of a controlled vector field.[6] Let be a system of tubular neighborhoods in of strata in where is the associated projection and given by the square norm on each fiber of . (The construction of such a system relies on the Whitney conditions or something weaker.) By definition, a controlled vector field is a family of vector fields (smooth of some class) on the strata such that: for each stratum A, there exists a neighborhood of in such that for any ,
on .
Assume the system is compatible with the map (such a system exists). Then there are two key results due to Thom:
- Given a vector field on N, there exists a controlled vector field on S that is a lift of it: .[7]
- A controlled vector field has a continuous flow (despite the fact that a controlled vector field is discontinuous).[8]
The lemma now follows in a straightforward fashion. Since the statement is local, assume and the coordinate vector fields on . Then, by the lifting result, we find controlled vector fields on such that . Let be the flows associated to them. Then define
by
It is a map over and is a homeomorphism since is the inverse. Since the flows preserve the strata, also preserves the strata.