Let x0 of type (ι → ι → ι) → (ι → ι) → ι be given.
Let x1 of type ((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) → ι) → ι) → ι → ι) → ι be given.
Let x3 of type (((ι → ι → ι) → ι → ι) → ι → ι) → ι → ι → ι → ι be given.
Assume H0:
∀ x4 x5 x6 x7 . x3 (λ x8 : (ι → ι → ι) → ι → ι . λ x9 . x8 (λ x10 x11 . x10) (Inj0 (x8 (λ x10 x11 . Inj0 0) (setsum 0 0)))) (x2 (λ x8 . λ x9 : ((ι → ι) → ι → ι) → ι → ι → ι . 0) (λ x8 : ((ι → ι) → ι) → ι . λ x9 . 0)) 0 (x1 (λ x8 : ι → ι . λ x9 : ι → (ι → ι) → ι → ι . λ x10 . setsum (Inj1 (x0 (λ x11 x12 . 0) (λ x11 . 0))) x10) x6) = setsum 0 0.
Assume H1: ∀ x4 : ι → ((ι → ι) → ι → ι) → ι . ∀ x5 x6 x7 . x3 (λ x8 : (ι → ι → ι) → ι → ι . λ x9 . x9) (x0 (λ x8 x9 . 0) (λ x8 . x8)) x6 (x0 (λ x8 x9 . 0) (λ x8 . 0)) = x6.
Assume H2:
∀ x4 : ι → ι . ∀ x5 . ∀ x6 : ι → (ι → ι → ι) → (ι → ι) → ι → ι . ∀ x7 . x2 (λ x8 . λ x9 : ((ι → ι) → ι → ι) → ι → ι → ι . 0) (λ x8 : ((ι → ι) → ι) → ι . λ x9 . Inj1 (x3 (λ x10 : (ι → ι → ι) → ι → ι . λ x11 . x10 (λ x12 x13 . x13) (x1 (λ x12 : ι → ι . λ x13 : ι → (ι → ι) → ι → ι . λ x14 . 0) 0)) 0 0 0)) = setsum (x0 (λ x8 x9 . setsum 0 (x6 (x2 (λ x10 . λ x11 : ((ι → ι) → ι → ι) → ι → ι → ι . 0) (λ x10 : ((ι → ι) → ι) → ι . λ x11 . 0)) (λ x10 x11 . x11) (λ x10 . 0) (setsum 0 0))) (λ x8 . setsum (Inj1 (Inj0 0)) (x1 (λ x9 : ι → ι . λ x10 : ι → (ι → ι) → ι → ι . λ x11 . setsum 0 0) x5))) x7.
Assume H3: ∀ x4 : (ι → (ι → ι) → ι) → ι → (ι → ι) → ι → ι . ∀ x5 . ∀ x6 : (((ι → ι) → ι) → (ι → ι) → ι) → ι → ι . ∀ x7 : ι → ((ι → ι) → ι) → ι . x2 (λ x8 . λ x9 : ((ι → ι) → ι → ι) → ι → ι → ι . 0) (λ x8 : ((ι → ι) → ι) → ι . λ x9 . x9) = x5.
Apply FalseE with
... ⟶ (∀ x4 . ∀ x5 : ι → (ι → ι) → ι → ι → ι . ∀ x6 . ∀ x7 : ι → ((ι → ι) → ι → ι) → ι . x1 (λ x8 : ι → ι . λ x9 : ι → (ι → ι) → ι → ι . λ x10 . x3 (λ x11 : (ι → ι → ι) → ι → ι . λ x12 . 0) 0 (Inj1 (setsum 0 ...)) ...) ... = ...) ⟶ (∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι . ∀ x7 : (((ι → ι) → ι → ι) → ι) → ι → ι . x0 (λ x8 x9 . Inj0 (Inj0 0)) (λ x8 . x3 (λ x9 : (ι → ι → ι) → ι → ι . λ x10 . x1 (λ x11 : ι → ι . λ x12 : ι → (ι → ι) → ι → ι . λ x13 . setsum x13 0) (setsum (setsum 0 0) 0)) 0 (Inj0 (Inj0 0)) (Inj0 (x2 (λ x9 . λ x10 : ((ι → ι) → ι → ι) → ι → ι → ι . x9) (λ x9 : ((ι → ι) → ι) → ι . λ x10 . 0)))) = Inj1 x4) ⟶ (∀ x4 x5 : ι → ι . ∀ x6 x7 . x0 (λ x8 x9 . x2 (λ x10 . λ x11 : ((ι → ι) → ι → ι) → ι → ι → ι . setsum 0 0) (λ x10 : ((ι → ι) → ι) → ι . λ x11 . 0)) (λ x8 . x1 (λ x9 : ι → ι . λ x10 : ι → (ι → ι) → ι → ι . λ x11 . 0) (Inj1 x7)) = x1 (λ x8 : ι → ι . λ x9 : ι → (ι → ι) → ι → ι . λ x10 . Inj0 (setsum (x3 (λ x11 : (ι → ι → ι) → ι → ι . λ x12 . x10) (Inj0 0) (x1 (λ x11 : ι → ι . λ x12 : ι → (ι → ι) → ι → ι . λ x13 . 0) 0) (x2 (λ x11 . λ x12 : ((ι → ι) → ι → ι) → ι → ι → ι . 0) (λ x11 : ((ι → ι) → ι) → ι . λ x12 . 0))) (setsum (setsum 0 0) (Inj1 0)))) (Inj1 (Inj0 (setsum (x5 0) (x0 (λ x8 x9 . 0) (λ x8 . 0)))))) ⟶ False.