  Let G be a Lie groupoid over M such that the target-source map from G to M x
M is proper. We show that, if O is an orbit of finite type (i.e. which admits a
proper function with finitely many critical points), then the restriction G|U
of G to some neighborhood U of O in M is isomorphic to a similar restriction of
the action groupoid for the linear action of the transitive groupoid G|O on the
normal bundle NO. The proof uses a deformation argument based on a cohomology
vanishing theorem, along with a slice theorem which is derived from a new
result on submersions with a fibre of finite type.
