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