Let x0 of type (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 be given.
Let x2 of type ((ι → 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 . ∀ x6 : ((ι → ι → ι) → ι) → ι . ∀ x7 . x3 (λ x8 x9 . x0 (λ x10 . λ x11 : ((ι → ι) → ι → ι) → ι . 0) (x0 (λ x10 . λ x11 : ((ι → ι) → ι → ι) → ι . Inj1 (Inj1 0)) (x2 (λ x10 : ι → ι . λ x11 : ι → (ι → ι) → ι → ι . x0 (λ x12 . λ x13 : ((ι → ι) → ι → ι) → ι . 0) 0) (λ x10 . x0 (λ x11 . λ x12 : ((ι → ι) → ι → ι) → ι . 0) 0)))) (x0 (λ x8 . λ x9 : ((ι → ι) → ι → ι) → ι . x9 (λ x10 : ι → ι . λ x11 . 0)) (Inj0 (setsum 0 x4))) = setsum x4 (Inj0 (Inj0 0)).
Assume H1: ∀ x4 x5 x6 : ι → ι . ∀ x7 . x3 (λ x8 x9 . x7) 0 = x7.
Assume H2:
∀ x4 x5 x6 . ∀ x7 : (((ι → ι) → ι → ι) → ι) → ((ι → ι) → ι) → (ι → ι) → ι . x2 (λ x8 : ι → ι . λ x9 : ι → (ι → ι) → ι → ι . x1 (λ x10 x11 . 0) 0 (λ x10 x11 x12 . x11) (λ x10 : ι → ι . λ x11 . x2 (λ x12 : ι → ι . λ x13 : ι → (ι → ι) → ι → ι . x11) (λ x12 . setsum 0 (setsum 0 0))) 0) (λ x8 . x0 (λ x9 . λ x10 : ((ι → ι) → ι → ι) → ι . x9) 0) = setsum 0 0.
Assume H3:
∀ x4 . ∀ x5 : ι → ι . ∀ x6 : ι → ι → ι . ∀ x7 : (((ι → ι) → ι → ι) → (ι → ι) → ι) → ι . x2 (λ x8 : ι → ι . λ x9 : ι → (ι → ι) → ι → ι . x3 (λ x10 x11 . 0) (x6 (setsum (setsum 0 0) (Inj1 0)) (x1 (λ x10 x11 . 0) (setsum 0 0) (λ x10 x11 x12 . x10) (λ x10 : ι → ι . λ x11 . Inj1 0) (Inj1 0)))) (λ x8 . x2 (λ x9 : ι → ι . λ x10 : ι → (ι → ι) → ι → ι . setsum (Inj1 (x2 (λ x11 : ι → ι . λ x12 : ι → (ι → ι) → ι → ι . 0) (λ x11 . 0))) 0) (λ x9 . x3 (λ x10 x11 . 0) 0)) = x3 (λ x8 x9 . x1 (λ x10 x11 . x10) (x0 (λ x10 . λ x11 : ((ι → ι) → ι → ι) → ι . x3 (λ x12 x13 . x1 (λ x14 x15 . 0) 0 (λ x14 x15 x16 . 0) (λ x14 : ι → ι . λ x15 . 0) 0) 0) (setsum (x0 (λ x10 . λ x11 : ((ι → ι) → ι → ι) → ι . 0) 0) (x6 0 0))) (λ x10 x11 x12 . 0) (λ x10 : ι → ι . λ x11 . x11) (x3 (λ x10 x11 . 0) (x1 (λ x10 x11 . setsum 0 0) 0 (λ x10 x11 x12 . setsum 0 0) (λ x10 : ι → ι . λ x11 . x11) (x2 (λ x10 : ι → ι . λ x11 : ι → (ι → ι) → ι → ι . 0) (λ x10 . 0))))) (setsum x4 (x3 (λ x8 x9 . 0) (Inj1 0))).
Assume H4:
∀ x4 : ((ι → ι) → ι) → (ι → ι → ι) → ι → ι . ∀ x5 : ι → ι → ι . ∀ x6 x7 . x1 (λ x8 x9 . x9) (Inj1 0) (λ x8 x9 x10 . x10) (λ x8 : ι → ι . λ x9 . 0) 0 = x6.
Apply FalseE with
... ⟶ ... ⟶ (∀ x4 : (((ι → ι) → ι → ι) → ι) → (ι → ι → ι) → (ι → ι) → ι . ∀ x5 : ((ι → ι) → ι) → ι . ∀ x6 x7 : ι → ι . x0 (λ x8 . λ x9 : ((ι → ι) → ι → ι) → ι . 0) (x6 (x5 (λ x8 : ι → ι . setsum (x8 0) (x5 (λ x9 : ι → ι . 0))))) = x6 (x1 (λ x8 x9 . x0 (λ x10 . λ x11 : ((ι → ι) → ι → ι) → ι . x0 (λ x12 . λ x13 : ((ι → ι) → ι → ι) → ι . setsum 0 0) (x1 (λ x12 x13 . 0) 0 ... ... 0)) ...) ... ... ... 0)) ⟶ False.