We denote by $d$ the degree of $\pi_f$, $E_1 = \operatorname{Vect}(x_1, f(x_1), \ldots, f^{d-1}(x_1))$, and $\psi_1$ is the endomorphism induced by $f$ on the vector subspace $E_1$,
$$\psi_1 : \left\lvert\, \begin{aligned} & E_1 \rightarrow E_1, \\ & x \mapsto f(x). \end{aligned} \right.$$
Justify that $\psi_1$ is cyclic.