0P7V Proof. Let τ∈Hom𝒮n(J,I)\tau\in\operatorname{Hom}\nolimits_{{\mathcal{S}}_{n}}(J,I) be an increasing bijection. We have D(τ∘σ)=D(σ)D(\tau\circ\sigma)=D(\sigma) and {σ′′∈End𝒮n(I)|σ′′<τ∘σ,ℓ(σ′′)=ℓ(τ∘σ)−1}={τ∘σ′|σ′∈Hom𝒮n(I,J),σ′<σ,ℓ(σ′)=ℓ(σ)−1}\{\sigma^{\prime\prime}\in\operatorname{End}\nolimits_{{\mathcal{S}}_{n}}(I)\ |\ \sigma^{\prime\prime}<\tau\circ\sigma,\ \ell(\sigma^{\prime\prime})=\ell(\tau\circ\sigma)-1\}=\{\tau\circ\sigma^{\prime}\ |\ \sigma^{\prime}\in\operatorname{Hom}\nolimits_{{\mathcal{S}}_{n}}(I,J),\ \sigma^{\prime}<\sigma,\ \ell(\sigma^{\prime})=\ell(\sigma)-1\} by Lemma 6.2.8. Since the first statement of the lemma holds for τ∘σ\tau\circ\sigma by Lemma 3.2.4, it holds for σ\sigma. The other statements follow from Lemmas 6.2.4 and 6.2.5. ∎