0P6N Proof. Let (m,ς)∈Δλ𝒲(m,\varsigma)\in\Delta_{\lambda}{\mathcal{W}}. We have E(m,π)=(m′,π′)E(m,\pi)=(m^{\prime},\pi^{\prime}) where m′=cone(π)m^{\prime}=\operatorname{cone}\nolimits(\pi) and π′\pi^{\prime} is given in §4.3.2. We have Γ∘E(m,ς)=(m′,π′′)\Gamma\circ E(m,\varsigma)=(m^{\prime},\pi^{\prime\prime}) where π12′′=E1E2ε1∘E1λE1∘η1E2E1=σ,π21′′=0\pi^{\prime\prime}_{12}=E_{1}E_{2}\varepsilon_{1}\circ E_{1}\lambda E_{1}\circ\eta_{1}E_{2}E_{1}=\sigma,\ \pi^{\prime\prime}_{21}=0 π11′′\displaystyle\pi^{\prime\prime}_{11} =E1E2ς1∘E1E2λ∘E1λE2∘E1F1τ2∘η1E22\displaystyle=E_{1}E_{2}\varsigma_{1}\circ E_{1}E_{2}\lambda\circ E_{1}\lambda E_{2}\circ E_{1}F_{1}\tau_{2}\circ\eta_{1}E_{2}^{2} =E1E2ς1∘E1E2λ∘E1λE2∘η1E22∘τ2\displaystyle=E_{1}E_{2}\varsigma_{1}\circ E_{1}E_{2}\lambda\circ E_{1}\lambda E_{2}\circ\eta_{1}E_{2}^{2}\circ\tau_{2} =E1E2ς1∘E1E2λ∘σF1E2∘E2η1E2∘τ2\displaystyle=E_{1}E_{2}\varsigma_{1}\circ E_{1}E_{2}\lambda\circ\sigma F_{1}E_{2}\circ E_{2}\eta_{1}E_{2}\circ\tau_{2} =σ∘E2E1ς1∘E2E1λ∘E2η1E2∘τ2\displaystyle=\sigma\circ E_{2}E_{1}\varsigma_{1}\circ E_{2}E_{1}\lambda\circ E_{2}\eta_{1}E_{2}\circ\tau_{2} =π11′\displaystyle=\pi^{\prime}_{11} π22′′\displaystyle\pi^{\prime\prime}_{22} =E12ς1∘E12λ∘E1ρE2∘E1F1σ∘η1E2E1\displaystyle=E_{1}^{2}\varsigma_{1}\circ E_{1}^{2}\lambda\circ E_{1}\rho E_{2}\circ E_{1}F_{1}\sigma\circ\eta_{1}E_{2}E_{1} =E12ς1∘E12λ∘E1ρE2∘η1E1E2∘σ\displaystyle=E_{1}^{2}\varsigma_{1}\circ E_{1}^{2}\lambda\circ E_{1}\rho E_{2}\circ\eta_{1}E_{1}E_{2}\circ\sigma =E12ς1∘E12λ∘τ1F1E2∘E1η1E2∘σ\displaystyle=E_{1}^{2}\varsigma_{1}\circ E_{1}^{2}\lambda\circ\tau_{1}F_{1}E_{2}\circ E_{1}\eta_{1}E_{2}\circ\sigma =τ1∘E12ς1∘E12λ∘E1η1E2∘σ\displaystyle=\tau_{1}\circ E_{1}^{2}\varsigma_{1}\circ E_{1}^{2}\lambda\circ E_{1}\eta_{1}E_{2}\circ\sigma =π22′\displaystyle=\pi^{\prime}_{22} It follows that π′′=π′\pi^{\prime\prime}=\pi^{\prime}. ∎