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