Let x0 of type (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) → ι → ((ι → ι) → ι → ι → ι) → ι be given.
Let x2 of type (ι → ι → ι) → ι → ι → ((ι → ι) → ι → ι) → ι be given.
Let x3 of type (ι → ι) → (ι → ι) → ι be given.
Assume H0:
∀ x4 . ∀ x5 : (ι → ι → ι → ι) → ι . ∀ x6 x7 . x3 (λ x8 . 0) (λ x8 . setsum (Inj1 0) (Inj1 0)) = x4.
Apply FalseE with
... ⟶ ... ⟶ ... ⟶ ... ⟶ (∀ x4 . ∀ x5 : ((ι → ι) → ι) → ((ι → ι) → ι → ι) → ι . ∀ x6 . ∀ x7 : (((ι → ι) → ι → ι) → ι) → ι → ι → ι . x1 (λ x8 : ι → ((ι → ι) → ι → ι) → ι → ι . Inj0 0) (x2 (λ x8 x9 . setsum (x2 (λ x10 x11 . x9) (x2 (λ x10 x11 . 0) 0 0 (λ x10 : ι → ι . λ x11 . 0)) (x7 (λ x10 : (ι → ι) → ι → ι . 0) 0 0) (λ x10 : ι → ι . λ x11 . x0 (λ x12 x13 . λ x14 : ι → ι → ι . λ x15 . 0) (λ x12 : ι → ι . 0) (λ x12 . 0))) (Inj1 (Inj0 0))) (x5 (λ x8 : ι → ι . 0) (λ x8 : ι → ι . λ x9 . x7 (λ x10 : (ι → ι) → ι → ι . Inj1 0) (x1 (λ x10 : ι → ((ι → ι) → ι → ι) → ι → ι . 0) 0 (λ x10 : ι → ι . λ x11 x12 . 0)) 0)) ... ...) ... = ...) ⟶ (∀ x4 : (ι → (ι → ι) → ι) → ι . ∀ x5 x6 . ∀ x7 : ((ι → ι → ι) → (ι → ι) → ι → ι) → ι . x0 (λ x8 x9 . λ x10 : ι → ι → ι . λ x11 . setsum 0 (x2 (λ x12 x13 . x13) x11 0 (λ x12 : ι → ι . λ x13 . Inj1 0))) (λ x8 : ι → ι . Inj0 x5) (λ x8 . 0) = Inj1 (x7 (λ x8 : ι → ι → ι . λ x9 : ι → ι . λ x10 . Inj1 (setsum x10 x10)))) ⟶ (∀ x4 : ι → ι → ι . ∀ x5 : (ι → (ι → ι) → ι → ι) → ι . ∀ x6 . ∀ x7 : ι → ((ι → ι) → ι) → ι . x0 (λ x8 x9 . λ x10 : ι → ι → ι . λ x11 . x10 (Inj1 (x2 (λ x12 x13 . setsum 0 0) (x3 (λ x12 . 0) (λ x12 . 0)) 0 (λ x12 : ι → ι . λ x13 . x2 (λ x14 x15 . 0) 0 0 (λ x14 : ι → ι . λ x15 . 0)))) (x2 (λ x12 x13 . x2 (λ x14 x15 . x0 (λ x16 x17 . λ x18 : ι → ι → ι . λ x19 . 0) (λ x16 : ι → ι . 0) (λ x16 . 0)) 0 0 (λ x14 : ι → ι . λ x15 . Inj1 0)) 0 (Inj0 (x1 (λ x12 : ι → ((ι → ι) → ι → ι) → ι → ι . 0) 0 (λ x12 : ι → ι . λ x13 x14 . 0))) (λ x12 : ι → ι . λ x13 . setsum x11 (setsum 0 0)))) (λ x8 : ι → ι . setsum (x2 (λ x9 x10 . x2 (λ x11 x12 . Inj0 0) x9 (x8 0) (λ x11 : ι → ι . λ x12 . Inj0 0)) x6 (Inj0 (x1 (λ x9 : ι → ((ι → ι) → ι → ι) → ι → ι . 0) 0 (λ x9 : ι → ι . λ x10 x11 . 0))) (λ x9 : ι → ι . λ x10 . x8 (Inj0 0))) (x8 (setsum (x3 (λ x9 . 0) (λ x9 . 0)) (setsum 0 0)))) (λ x8 . Inj0 (x7 0 (λ x9 : ι → ι . x6))) = setsum (Inj0 (x5 (λ x8 . λ x9 : ι → ι . λ x10 . x7 (x2 (λ x11 x12 . 0) 0 0 (λ x11 : ι → ι . λ x12 . 0)) (λ x11 : ι → ι . x0 (λ x12 x13 . λ x14 : ι → ι → ι . λ x15 . 0) (λ x12 : ι → ι . 0) (λ x12 . 0))))) (setsum (x3 (λ x8 . 0) (λ x8 . setsum x6 (Inj1 0))) 0)) ⟶ False.