Let $P_{\mathcal{Y}^{\perp},A\mathcal{Z}}$ be a projection with range $\mathcal{Y}^{\perp}$ and kernel $A\mathcal{Z}$, i.e. $P_{\mathcal{Y}^{\perp},A\mathcal{Z}}=I-AZ(Y^HAZ)^{-1}Y^H$, where $\mathbb{C}^n=A\mathcal{Z} \oplus \mathcal{Y}^{\perp}$ and $Z$, $Y$ are matrices which columns form a basis of $\mathcal{Y}$ and $\mathcal{Z}$. Furthermore, we define the search space $\mathcal{S}_i=K_i(P_{\mathcal{Y}^{\perp},A\mathcal{Z}}A,P_{\mathcal{Y}^{\perp},A\mathcal{Z}}r_0)$, where $r_0=b-Ax_0$.
Now the paper states that from $\mathcal{S}_i \cap \mathcal{N}(P_{\mathcal{Y}^{\perp},A\mathcal{Z}}A)=\{0\}$ follows that $\mathcal{S}_i \subset \mathcal{Y}^{\perp}$ and $\mathcal{S}_i \cap \mathcal{N}(P_{\mathcal{Y}^{\perp},A\mathcal{Z}}A)=\mathcal{S}_i \cap \mathcal{Z}$. Why?