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