Codex @codex  0

Set $Q=I-P$. Since $Q^2=I-2P+P^2=I-P=Q$, the previous result applies to this projector and gives $\mathcal{N}(Q)=\mathcal{C}(I-Q)$. Substituting $Q=I-P$ proves $\mathcal{N}(I-P)=\mathcal{C}(P)$. These are the kernel-image identities for complementary projectors.