Proof of proposition 9.12 in Axler

Question

The proof started out with assuming
$$ (T_C-\lambda I)^j(u+iv)=(T_C-\lambda I)^{j-1}((T_C-\lambda I)(u+iv))=0, $$ where $T_C$ is the complexification of $T \in \mathcal L(V)$. So, $$ (T_C-\lambda I)(u+iv)=(Tu-au+bv)+i(Tv-av-bu) $$ and $$ (T_C-\bar{\lambda} I)(u+iv)=(Tu-au+bv)-i(Tv-av-bu), $$ where $\lambda=a+bi$. The part I don't understand is the proof proceeds to imply that $$ (T_C-\bar{\lambda} I)^{j-1}((Tu-au+bv)-i(Tv-av-bu))=0. $$ Why is that?