Lemma 1 of page 35 says that the index at an isolated zero z of vector field v on an open set U of R^m is the same as the index at f(z) of the pushfoward f_*v = df o v o f^-1 of v by a diffeomorphism f:U-->U'. For the proof, he first reduces the problem to the case where z=f(z)=0 and is U a small open ball. Then he proves the lemma in the case where f preserves the orientation. Then he arrives to the case when f reverses it and says, "To consider diffeomorphisms which reverse orientation, it is sufficient to consider the special case of a reflection." Why is this sufficient?!?