we denote $\lambda$. This transforms $A^{\sigma,\pi}$ into the interval $(\sigma,\pi)$. We need to know what effect this has on the Möbius function of $A^{\sigma,\pi}$.

We work our way from the bottom to the top so we can assume that all elements below the elements of $\widehat{E}^{\lambda,\pi}$ have already been retracted and all elements above have not. Define the poset $W(\lambda):=\{\tau\in A^{\sigma,\pi}\,|\,\tau\le\eta\text{ or } \tau\ge\eta\text{ for some } \eta\in\widehat{E}^{\lambda,\pi}\}$. When we retract the elements of $\widehat{E}^{\lambda,\pi}$ to $\lambda$ we retract $W(\lambda)$ onto a contractible poset, since in that poset the element $\lambda$ is comparable to all other elements and thus represents a cone point in the corresponding order complex. This implies the change to the Möbius function is $-\mu(W(\lambda))$.

To compute $\mu(W(\lambda))$ we split $W(\lambda)$ into two disjoint parts $$\begin{aligned}
W(\lambda)^<&:=\{\tau\in W(\lambda)\,|\,\tau<\eta\text{ for some } \eta\in\widehat{E}^{\lambda,\pi}\},\\W(\lambda)^\ge&:=\{\tau\in W(\lambda)\,|\,\tau\ge\eta\text{ for some } \eta\in\widehat{E}^{\lambda,\pi}\}.\end{aligned}$$ The poset $W(\lambda)^<$ is isomorphic to $(\sigma,\lambda)$ because all points below $\lambda$ have already been retracted. The poset $W(\lambda)^\ge$ is equal to $\bigcup_{\eta\in\widehat{E}^{\lambda,\pi}}(P(\eta)\setminus\hat{\pi})$ which has Möbius function $-\sum_{S\in EZ^{\lambda,\pi}}(-1)^{|S|}$, by Lemma \[lem:inter\] and the inclusion-exclusion formula (this also follows from the Crosscut Theorem, see Proposition \[prop:crosscut\]).

Because every element of $\widehat{E}^{\lambda,\pi}$ lies above every element of $(\sigma,\lambda)$ this implies $W(\lambda)=W(\lambda)^<\star 