Let x0 of type (ι → (ι → ι) → (ι → ι) → ι) → ι → ι be given.
Let x1 of type (ι → ι) → ((((ι → ι) → ι → ι) → ι → ι) → ι → (ι → ι) → ι) → ι be given.
Let x2 of type (ι → ι) → ι → ((ι → ι) → ι → ι → ι) → ι be given.
Let x3 of type (ι → ι) → ((((ι → ι) → ι) → ι) → ι) → ι be given.
Assume H0:
∀ x4 : (((ι → ι) → ι → ι) → (ι → ι) → ι) → ι . ∀ x5 : ((ι → ι → ι) → ι → ι) → (ι → ι) → ι . ∀ x6 x7 . x3 (λ x8 . setsum x8 (x5 (λ x9 : ι → ι → ι . λ x10 . x2 (λ x11 . x10) x10 (λ x11 : ι → ι . λ x12 x13 . x10)) (λ x9 . 0))) (λ x8 : ((ι → ι) → ι) → ι . x8 (λ x9 : ι → ι . 0)) = Inj1 (Inj0 x7).
Assume H1:
∀ x4 : ((ι → ι → ι) → ι → ι) → ι → (ι → ι) → ι → ι . ∀ x5 . ∀ x6 : ι → ι . ∀ x7 : ι → (ι → ι) → (ι → ι) → ι → ι . x3 (λ x8 . 0) (λ x8 : ((ι → ι) → ι) → ι . Inj1 0) = x7 (Inj0 (x4 (λ x8 : ι → ι → ι . λ x9 . 0) (setsum 0 (x1 (λ x8 . 0) (λ x8 : ((ι → ι) → ι → ι) → ι → ι . λ x9 . λ x10 : ι → ι . 0))) (λ x8 . x8) 0)) (λ x8 . x7 (x2 (λ x9 . 0) 0 (λ x9 : ι → ι . λ x10 x11 . 0)) (λ x9 . x9) (λ x9 . x3 (λ x10 . setsum 0 (x3 (λ x11 . 0) (λ x11 : ((ι → ι) → ι) → ι . 0))) (λ x10 : ((ι → ι) → ι) → ι . x1 (λ x11 . 0) (λ x11 : ((ι → ι) → ι → ι) → ι → ι . λ x12 . λ x13 : ι → ι . x2 (λ x14 . 0) 0 (λ x14 : ι → ι . λ x15 x16 . 0)))) 0) (λ x8 . x3 (λ x9 . Inj0 (x6 (setsum 0 0))) (λ x9 : ((ι → ι) → ι) → ι . 0)) (Inj1 (x2 (λ x8 . x7 0 (λ x9 . x6 0) (λ x9 . x2 (λ x10 . 0) 0 (λ x10 : ι → ι . λ x11 x12 . 0)) (Inj1 0)) x5 (λ x8 : ι → ι . λ x9 x10 . x1 (λ x11 . setsum 0 0) (λ x11 : ((ι → ι) → ι → ι) → ι → ι . λ x12 . λ x13 : ι → ι . x1 (λ x14 . 0) (λ x14 : ((ι → ι) → ι → ι) → ι → ι . λ x15 . λ x16 : ι → ι . 0))))).
Apply FalseE with
... ⟶ ... ⟶ ... ⟶ (∀ x4 . ∀ x5 : ι → ι . ∀ x6 x7 . x1 (λ x8 . x1 (λ x9 . x0 (λ x10 . λ x11 x12 : ι → ι . Inj1 0) (x0 (λ x10 . λ x11 x12 : ι → ι . 0) (x0 (λ x10 . λ x11 x12 : ι → ι . 0) 0))) (λ x9 : ((ι → ι) → ι → ι) → ι → ι . λ x10 . λ x11 : ι → ι . Inj1 (x2 (λ x12 . x2 (λ x13 . 0) 0 (λ x13 : ι → ι . λ x14 x15 . 0)) (x0 (λ x12 . λ x13 x14 : ι → ι . 0) 0) (λ x12 : ι → ι . λ x13 x14 . x3 (λ x15 . 0) (λ x15 : ((ι → ι) → ι) → ι . 0))))) (λ x8 : ((ι → ι) → ι → ι) → ι → ι . λ x9 . λ x10 : ι → ι . x0 (λ x11 . λ x12 x13 : ι → ι . x12 x11) (x8 (λ x11 : ι → ι . λ x12 . x10 0) (x2 (λ x11 . x3 (λ x12 . 0) (λ x12 : ((ι → ι) → ι) → ι . 0)) ... ...))) = ...) ⟶ (∀ x4 : (((ι → ι) → ι → ι) → ι) → ι → ι . ∀ x5 : (((ι → ι) → ι → ι) → ι → ι) → ι . ∀ x6 . ∀ x7 : ((ι → ι) → ι) → ι . x0 (λ x8 . λ x9 x10 : ι → ι . 0) 0 = x4 (λ x8 : (ι → ι) → ι → ι . x6) (setsum (Inj0 (x0 (λ x8 . λ x9 x10 : ι → ι . x3 (λ x11 . 0) (λ x11 : ((ι → ι) → ι) → ι . 0)) (x5 (λ x8 : (ι → ι) → ι → ι . λ x9 . 0)))) (x0 (λ x8 . λ x9 x10 : ι → ι . setsum 0 (setsum 0 0)) x6))) ⟶ (∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 . ∀ x7 : (ι → ι) → ι . x0 (λ x8 . λ x9 x10 : ι → ι . x8) x6 = x6) ⟶ False.