We denote by $d$ the degree of $\pi_f$, $E_1 = \operatorname{Vect}(e_1, e_2, \ldots, e_d)$ where $e_i = f^{i-1}(x_1)$, $F = \{x \in E \mid \forall i \in \mathbb{N}, \Phi(f^i(x)) = 0\}$, and $\Psi$ is the linear map from $E$ to $\mathbb{K}^d$ defined by $\Psi(x) = \left(\Phi(f^i(x))\right)_{0 \leqslant i \leqslant d-1}$.
Show that $E = E_1 \oplus F$.