Greetings, I'm trying to understand a theorem about partial feedback linearization from a paper "On the largest feedback linearizable subsystem" by R.Marino (you can find it here: http://www.mediafire.com/?si2t4lnhxa4jt82)

My question is concerning proof of theorem 4. Citing:

Consider $\overline{G}^{\overline{k}^*_1-2}$. It is easy to see that there must exist an $(r_{k^*_1-1})$-vector function $\phi$, such that $$ d\phi_1 \subset (\overline{G}^{k^*_1-2})^{\bot} $$ and $$ rank <d \phi_1, ad _f^{\overline{k}^*_1-1} G>=r_{\overline{k}^*_1-1} $$

Is the first proposition a consequence of Frobenius theorem? Where does the second proposition come from?