\begin{lemma}On the infinite chain

		$$\P_I(Z^{0}\cap Z^{k})=\P_I(Z^{0})\P_I(Z^{k})+\mathcal{O}(p^{k-|b|+1})$$

	\end{lemma}   

    Note that using De Morgan's law and the inclusion-exclusion formula we can see that this is equivalent to saying:

    $$\P_I(NZ^{0}\cap NZ^{k})=\P_I(NZ^{0})\P_I(NZ^{k})+\mathcal{O}(p^{k-|b|+1})$$

    \begin{proof}

    We proceed by induction on $|I|$. For $|I|=0,1$ the statement is trivial.

    Now observe that:

    $$\P_I(Z^{0})=\sum_{P\text{ patch}\,:\,0\in P}\P_I(P\in\mathcal{P})$$

    $$\P_I(Z^{k})=\sum_{P\text{ patch}\,:\,k\in P}\P_I(P\in\mathcal{P})$$

    Suppose $|I|\geq 2$, then we proceed using induction similarly to the above

    \begin{align*}

    &\P_I(Z^{0}\cap Z^{k})\\

    &=\sum_{\underset{P_l^{\max}+1<P_r^{\min}}{P_l,P_r\text{ patches}\,:\,0\in P_l,k\in P_r}}

    \P_I(P_l,P_r\in\mathcal{P})

    +\sum_{P\text{ patch}\,:\,0,k\in P}\P_I(P\in\mathcal{P})\\

    &=\sum_{\underset{P_l^{\max}+1<P_r^{\min}}{P_l,P_r\text{ patches}\,:\,0\in P_l,k\in P_r}}

    \P_I(P_l,P_r\in\mathcal{P})

    +\mathcal{O}(p^{k+1})\\

    &\overset{Lemma~\ref{claim:eventindependence}}{=}\sum_{\underset{P_l^{\max}+1<P_r^{\min}}{P_l,P_r\text{ patches}\,:\,0\in P_l,k\in P_r}}

    \P_{I\cap P_l}(P_l\in\mathcal{P})

    \P_{> I <}(NZ^{P_l^{\max}+1}\cap NZ^{P_r^{\min}-1})

    \P_{I\cap P_r}(P_r\in\mathcal{P})

    +\mathcal{O}(p^{k+1})\\

    &\overset{\text{induction}}{=}\sum_{\underset{P_l^{\max}+1<P_r^{\min}}{P_l,P_r\text{ patches}\,:\,0\in P_l,k\in P_r}}

    \P_{I\cap P_l}(P_l\in\mathcal{P})

    \P_{> I <}(NZ^{P_l^{\max}+1})\P_{> I <}(NZ^{P_r^{\min}-1})

    \P_{I\cap P_r}(P_r\in\mathcal{P})

    +\mathcal{O}(p^{k-|I|+1})\\

    &\overset{Lemma~\ref{lemma:probIndep}}{=}\sum_{\underset{P_l^{\max}+1<P_r^{\min}}{P_l,P_r\text{ patches}\,:\,0\in P_l,k\in P_r}}

    \P_{I\cap P_l}(P_l\in\mathcal{P})

    \P_{I\setminus P_l}(NZ^{P_l^{\max}+1})\P_{I\setminus P_r}(NZ^{P_r^{\min}-1})

    \P_{I\cap P_r}(P_r\in\mathcal{P})

    +\mathcal{O}(p^{k-|I|+1})\\

    &\overset{Lemma~\ref{claim:eventindependence}}{=}\sum_{\underset{P_l^{\max}+1<P_r^{\min}}{P_l,P_r\text{ patches}\,:\,0\in P_l,k\in P_r}}

    \P_{I}(P_l\in\mathcal{P})

    \P_{I}(P_r\in\mathcal{P})

    +\mathcal{O}(p^{k-|I|+1})\\

    &=\left(\sum_{P_l\text{ patch}\,:\,0\in P_l}

    \P_{I}(P_l\in\mathcal{P})\right)\left(\sum_{P_l\text{ patch}\,:\,0\in P_l}

    \P_{I}(P_l\in\mathcal{P})\right)

    +\mathcal{O}(p^{k-|I|+1})\\

    &=\P_I(Z^{0})\P_I(Z^{k})

    +\mathcal{O}(p^{k-|I|+1}).	

    \end{align*}

    \end{proof}

	Final observation: Suppose $S={a,b}$ 

	    \begin{align*}

	    &\sum_{f_{\overline{P}}\in\{0,1'\}^{2}} \rho_{S(f_{\overline{P}})} \mathbb{P}_{S(f_{\overline{P}})}(NZ^{(P_{\min}-1)}\cap NZ^{(P_{\max}+1)})  \\

	    &= \sum_{f_{\overline{P}}\in\{0,1'\}^{2}} \rho_{S(f_{\overline{P}})} \mathbb{P}_{S(f_{\overline{P}})}(NZ^{(P_{\min}-1)}) \mathbb{P}_{S(f_{\overline{P}})}(NZ^{(P_{\max}+1)})   +\mathcal{O}(p^{|\overline{P}|-|S|})\\

 	    &= \sum_{f_{\overline{P}}\in\{0,1'\}^{2}}  \rho_{a_f}\mathbb{P}_{S(f_{\overline{P}})}(NZ^{(P_{\min}-1)})  \rho_{b_f}\mathbb{P}_{S(f_{\overline{P}})}(NZ^{(P_{\max}+1)})   +\mathcal{O}(p^{|\overline{P}|-|S|})\\

	    \end{align*}