I'm having trouble understanding a lemma presented in Katok & Hasselblatt Introduction to the modern theory of dynamical systems (p.247). Here's the said lemma :
I don't understand the second part of the proof. More specifically, how does it come that if $(u,v) \in \tilde{V}^\gamma_p$, then $Df_m(u,v) = (u',v') \in V^\gamma_{f_m(p)}$ ?
By definition, we have that $(u,v) \in (Df_{m-1})_{f_{m-1}^{-1}(p)} V^\gamma_{f_{m-1}^{-1}(p)}$, but I don't see why this implies the conclusion...
Any help is greatly appreciated !