Let x0 of type ((((ι → ι) → ι) → (ι → ι → ι) → ι → ι) → ((ι → ι → ι) → (ι → ι) → ι → ι) → ι → ι → ι → ι) → ι → (ι → ι) → ι → ι be given.
Let x1 of type ((ι → CN (ι → ι)) → ι → ι) → (ι → ι) → ι be given.
Let x2 of type (ι → ι) → ι → ι be given.
Let x3 of type (ι → ι → ((ι → ι) → ι → ι) → (ι → ι) → ι) → (ι → ι) → ι be given.
Assume H0:
∀ x4 : (((ι → ι) → ι → ι) → ι → ι → ι) → ι → ι . ∀ x5 x6 x7 . x3 (λ x8 x9 . λ x10 : (ι → ι) → ι → ι . λ x11 : ι → ι . setsum (x0 (λ x12 : ((ι → ι) → ι) → (ι → ι → ι) → ι → ι . λ x13 : (ι → ι → ι) → (ι → ι) → ι → ι . λ x14 x15 x16 . x2 (λ x17 . Inj0 0) (x3 (λ x17 x18 . λ x19 : (ι → ι) → ι → ι . λ x20 : ι → ι . 0) (λ x17 . 0))) (x2 (λ x12 . Inj1 0) (x10 (λ x12 . 0) 0)) (λ x12 . x3 (λ x13 x14 . λ x15 : (ι → ι) → ι → ι . λ x16 : ι → ι . x13) (λ x13 . 0)) (Inj0 (x3 (λ x12 x13 . λ x14 : (ι → ι) → ι → ι . λ x15 : ι → ι . 0) (λ x12 . 0)))) (x3 (λ x12 x13 . λ x14 : (ι → ι) → ι → ι . λ x15 : ι → ι . 0) (λ x12 . x10 (λ x13 . setsum 0 0) (Inj1 0)))) (λ x8 . 0) = Inj0 0.
Assume H1:
∀ x4 : (ι → ι → ι) → (ι → ι) → ι . ∀ x5 x6 x7 . x3 (λ x8 x9 . λ x10 : (ι → ι) → ι → ι . λ x11 : ι → ι . x3 (λ x12 x13 . λ x14 : (ι → ι) → ι → ι . λ x15 : ι → ι . 0) (λ x12 . Inj0 x9)) (λ x8 . setsum x8 0) = x3 (λ x8 x9 . λ x10 : (ι → ι) → ι → ι . λ x11 : ι → ι . x3 (λ x12 x13 . λ x14 : (ι → ι) → ι → ι . λ x15 : ι → ι . x14 (λ x16 . 0) x12) (λ x12 . x3 (λ x13 x14 . λ x15 : (ι → ι) → ι → ι . λ x16 : ι → ι . 0) (λ x13 . 0))) (λ x8 . x8).
Assume H2: ∀ x4 : (ι → ι) → ι . ∀ x5 x6 x7 . x2 (λ x8 . 0) 0 = x7.
Apply FalseE with
... ⟶ (∀ x4 : (ι → ι) → ι → ι . ∀ x5 : ι → ι → ι . ∀ x6 : (ι → (ι → ι) → ι) → ι → ι . ∀ x7 . x1 (λ x8 : ι → ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . λ x9 . Inj0 (x3 (λ x10 x11 . λ x12 : (ι → ι) → ι → ι . λ x13 : ι → ι . x13 (setsum 0 0)) ...)) ... = ...) ⟶ (∀ x4 x5 . ∀ x6 : ι → (ι → ι → ι) → ι . ∀ x7 . x1 (λ x8 : ι → ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . λ x9 . x7) (λ x8 . 0) = x7) ⟶ (∀ x4 : ι → ι . ∀ x5 : (ι → ι → ι) → ι → (ι → ι) → ι . ∀ x6 : (ι → ι) → ι . ∀ x7 . x0 (λ x8 : ((ι → ι) → ι) → (ι → ι → ι) → ι → ι . λ x9 : (ι → ι → ι) → (ι → ι) → ι → ι . λ x10 x11 x12 . x11) (x4 (setsum 0 x7)) (λ x8 . x7) (setsum 0 (x4 (x4 (x0 (λ x8 : ((ι → ι) → ι) → (ι → ι → ι) → ι → ι . λ x9 : (ι → ι → ι) → (ι → ι) → ι → ι . λ x10 x11 x12 . 0) 0 (λ x8 . 0) 0)))) = x4 (setsum 0 (x4 (x0 (λ x8 : ((ι → ι) → ι) → (ι → ι → ι) → ι → ι . λ x9 : (ι → ι → ι) → (ι → ι) → ι → ι . λ x10 x11 x12 . 0) (setsum 0 0) (λ x8 . x2 (λ x9 . 0) 0) (x3 (λ x8 x9 . λ x10 : (ι → ι) → ι → ι . λ x11 : ι → ι . 0) (λ x8 . 0)))))) ⟶ (∀ x4 : (ι → ι → ι) → (ι → ι → ι) → ι . ∀ x5 . ∀ x6 x7 : ι → ι . x0 (λ x8 : ((ι → ι) → ι) → (ι → ι → ι) → ι → ι . λ x9 : (ι → ι → ι) → (ι → ι) → ι → ι . λ x10 x11 x12 . Inj1 (setsum (Inj0 (x2 (λ x13 . 0) 0)) (x2 (λ x13 . 0) (setsum 0 0)))) 0 (λ x8 . x7 (x3 (λ x9 x10 . λ x11 : (ι → ι) → ι → ι . λ x12 : ι → ι . setsum (x3 (λ x13 x14 . λ x15 : (ι → ι) → ι → ι . λ x16 : ι → ι . 0) (λ x13 . 0)) (x12 0)) (λ x9 . x1 (λ x10 : ι → ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . λ x11 . Inj1 0) (λ x10 . 0)))) 0 = setsum (setsum 0 (x4 (λ x8 x9 . 0) (λ x8 x9 . setsum (Inj1 0) 0))) 0) ⟶ False.