Let x0 of type ((ι → ι) → ((ι → 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) → ι) → ι → ι be given.
Let x2 of type (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 → ι) → ι be given.
Assume H0:
∀ x4 : (ι → ι → ι) → ι . ∀ x5 . ∀ x6 : (((ι → ι) → ι) → (ι → ι) → ι → ι) → ι → ι . ∀ x7 : ((ι → ι → ι) → ι) → (ι → ι) → ι → ι → ι . x3 (λ x8 : ((ι → ι → ι) → (ι → ι) → ι) → ι → ι → ι . setsum (x0 (λ x9 : ι → ι . λ x10 : (ι → ι → ι) → ι → ι . λ x11 . λ x12 : ι → ι . x1 (λ x13 . λ x14 : ((ι → ι) → ι → ι) → ι → ι → ι . setsum 0 0) (Inj1 0)) (λ x9 x10 . λ x11 : ι → ι . 0)) (x1 (λ x9 . λ x10 : ((ι → ι) → ι → ι) → ι → ι → ι . 0) (x2 (λ x9 x10 . x7 (λ x11 : ι → ι → ι . 0) (λ x11 . 0) 0 0) (λ x9 . 0) (x6 (λ x9 : (ι → ι) → ι . λ x10 : ι → ι . λ x11 . 0) 0) (x1 (λ x9 . λ x10 : ((ι → ι) → ι → ι) → ι → ι → ι . 0) 0) (Inj0 0)))) (λ x8 . x8) = Inj1 0.
Assume H1:
∀ x4 x5 . ∀ x6 : ι → ι . ∀ x7 . x3 (λ x8 : ((ι → ι → ι) → (ι → ι) → ι) → ι → ι → ι . 0) (λ x8 . x6 x8) = x6 (setsum (x2 (λ x8 x9 . x1 (λ x10 . λ x11 : ((ι → ι) → ι → ι) → ι → ι → ι . 0) (setsum 0 0)) (λ x8 . x8) x5 0 0) (x1 (λ x8 . λ x9 : ((ι → ι) → ι → ι) → ι → ι → ι . x9 (λ x10 : ι → ι . λ x11 . 0) x8 (x2 (λ x10 x11 . 0) (λ x10 . 0) 0 0 0)) 0)).
Assume H2:
∀ x4 . ∀ x5 : (ι → (ι → ι) → ι) → ι . ∀ x6 . ∀ x7 : ι → ι . x2 (λ x8 x9 . 0) (λ x8 . x2 (λ x9 x10 . x2 (λ x11 x12 . 0) (λ x11 . 0) (x0 (λ x11 : ι → ι . λ x12 : (ι → ι → ι) → ι → ι . λ x13 . λ x14 : ι → ι . setsum 0 0) (λ x11 x12 . λ x13 : ι → ι . x0 (λ x14 : ι → ι . λ x15 : (ι → ι → ι) → ι → ι . λ x16 . λ x17 : ι → ι . 0) (λ x14 x15 . λ x16 : ι → ι . 0))) (x7 x10) x10) (λ x9 . x8) (x1 (λ x9 . λ x10 : ((ι → ι) → ι → ι) → ι → ι → ι . 0) (x2 (λ x9 x10 . Inj1 0) (λ x9 . x1 (λ x10 . λ x11 : ((ι → ι) → ι → ι) → ι → ι → ι . 0) 0) (x3 (λ x9 : ((ι → ι → ι) → (ι → ι) → ι) → ι → ι → ι . 0) (λ x9 . 0)) (Inj1 0) (x0 (λ x9 : ι → ι . λ x10 : (ι → ι → ι) → ι → ι . λ x11 . λ x12 : ι → ι . 0) (λ x9 x10 . λ x11 : ι → ι . 0)))) (x7 0) 0) x4 (x0 (λ x8 : ι → ι . λ x9 : (ι → ι → ι) → ι → ι . λ x10 . λ x11 : ι → ι . Inj0 0) (λ x8 x9 . λ x10 : ι → ι . Inj0 (setsum (Inj0 0) (x1 (λ x11 . λ x12 : ((ι → ι) → ι → ι) → ι → ι → ι . 0) 0)))) (setsum 0 (x7 0)) = x4.
Assume H3:
∀ x4 : ι → ι → ι . ∀ x5 : (((ι → ι) → ι) → (ι → ι) → ι → ι) → ι → ι → ι → ι . ∀ x6 : ((ι → ι → ι) → ι → ι) → ι . ∀ x7 : ι → ι . x2 (λ x8 x9 . x2 (λ x10 x11 . 0) (λ x10 . x8) (x7 0) (x6 (λ x10 : ι → ι → ι . λ x11 . setsum 0 (x0 (λ x12 : ι → ι . λ x13 : (ι → ι → ι) → ι → ι . λ x14 . λ x15 : ι → ι . 0) (λ x12 x13 . λ x14 : ι → ι . 0)))) 0) (λ x8 . 0) (x5 (λ x8 : (ι → ι) → ι . λ x9 : ι → ι . λ x10 . 0) 0 (x2 (λ x8 x9 . x3 (λ x10 : ((ι → ι → ι) → (ι → ι) → ι) → ι → ι → ι . x9) (λ x10 . Inj0 0)) (λ x8 . x3 (λ x9 : ((ι → ι → ι) → (ι → ι) → ι) → ι → ι → ι . x6 (λ x10 : ι → ι → ι . λ x11 . 0)) (λ x9 . Inj0 0)) ... 0 ...) ...) ... ... = ....