 For Cut-inductivity suppose (1) Γ, Γ′, φ → ψ ≻ Δ, Δ′ has arisen by (→ Left) from (1a) Γ ≻ φ, Δ and (2a) Γ′, ψ ≻ Δ′, while (2) Γ′′ ≻ φ → ψ, Δ′′ has arisen by (→ Right) from (2a) Γ′′, φ ≻ ψ, Δ′′. We need to show that a cut on (1) and (2) with cut-formula φ → ψ can be avoided on the hypothesis that we can help ourselves to cuts with cut-formulas φ, ψ. From (1a) and (2a), cutting on φ, we get (3) Γ, Γ′′ ≻ ψ, Δ′′. Then cutting on ψ, (2b) and (3) deliver the desired conclusion: Γ, Γ′, Γ′′ ≻ Δ, Δ′, Δ′′, without the need to use φ → ψ as a cut-formula with (1) and (2) as premisses.