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