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 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) → 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 → ι) → ι → ι be given.
Assume H0:
∀ x4 . ∀ x5 : (ι → ι) → ι . ∀ x6 : ι → (ι → ι) → ι . ∀ x7 : ((ι → ι) → ι → ι → ι) → (ι → ι → ι) → ι → ι . x3 (λ x8 : (((ι → ι) → ι → ι) → ι) → ((ι → ι) → ι → ι) → ι . setsum 0 (setsum 0 0)) (λ x8 x9 x10 x11 . x0 (λ x12 : (ι → ι) → ι . x12 (λ x13 . 0)) (λ x12 : ι → ι . 0) (λ x12 : (ι → ι) → ι . x11)) (x3 (λ x8 : (((ι → ι) → ι → ι) → ι) → ((ι → ι) → ι → ι) → ι . 0) (λ x8 x9 x10 x11 . x8) 0) = setsum (x5 (λ x8 . 0)) x4.
Assume H1:
∀ x4 . ∀ x5 : (ι → ι) → ι → (ι → ι) → ι . ∀ x6 : ι → ι . ∀ x7 . x3 (λ x8 : (((ι → ι) → ι → ι) → ι) → ((ι → ι) → ι → ι) → ι . 0) (λ x8 x9 x10 x11 . x9) (setsum (Inj1 (x5 (λ x8 . x1 (λ x9 . 0) (λ x9 . 0) 0 0 (λ x9 . 0)) (x1 (λ x8 . 0) (λ x8 . 0) 0 0 (λ x8 . 0)) (λ x8 . 0))) (setsum (x6 (Inj1 0)) (x0 (λ x8 : (ι → ι) → ι . setsum 0 0) (λ x8 : ι → ι . 0) (λ x8 : (ι → ι) → ι . setsum 0 0)))) = setsum (setsum x4 (x6 (Inj0 (x5 (λ x8 . 0) 0 (λ x8 . 0))))) (Inj0 0).
Assume H2:
∀ x4 . ∀ x5 x6 : ι → ι . ∀ x7 . x2 (λ x8 : ι → (ι → ι) → ι . λ x9 : ι → ι . λ x10 : (ι → ι) → ι → ι . Inj1 (setsum (x8 (x8 0 (λ x11 . 0)) (λ x11 . x8 0 (λ x12 . 0))) (Inj1 (x2 (λ x11 : ι → (ι → ι) → ι . λ x12 : ι → ι . λ x13 : (ι → ι) → ι → ι . 0) 0 (λ x11 . λ x12 : ι → ι . λ x13 . 0) (λ x11 : ι → ι . λ x12 . 0) 0)))) 0 (λ x8 . λ x9 : ι → ι . λ x10 . 0) (λ x8 : ι → ι . λ x9 . x2 (λ x10 : ι → (ι → ι) → ι . λ x11 : ι → ι . λ x12 : (ι → ι) → ι → ι . 0) (Inj1 0) (λ x10 . λ x11 : ι → ι . λ x12 . setsum x10 (x11 x10)) (λ x10 : ι → ι . λ x11 . x2 (λ x12 : ι → (ι → ι) → ι . λ x13 : ι → ι . λ x14 : (ι → ι) → ι → ι . Inj1 (Inj1 0)) (x10 (setsum 0 0)) (λ x12 . λ x13 : ι → ι . λ x14 . x2 (λ x15 : ι → (ι → ι) → ι . λ x16 : ι → ι . λ x17 : (ι → ι) → ι → ι . x0 (λ x18 : (ι → ι) → ι . 0) (λ x18 : ι → ι . 0) (λ x18 : (ι → ι) → ι . 0)) (Inj0 0) (λ x15 . λ x16 : ι → ι . λ x17 . setsum 0 0) (λ x15 : ι → ι . λ x16 . x15 0) 0) (λ x12 : ι → ι . λ x13 . x11) 0) (Inj0 0)) x7 = Inj1 0.
Assume H3:
∀ x4 : ((ι → ι → ι) → ι) → ι . ∀ x5 x6 x7 . x2 (λ x8 : ι → (ι → ι) → ι . λ x9 : ι → ι . λ x10 : (ι → ι) → ι → ι . 0) (x4 (λ x8 : ι → ι → ι . x1 (λ x9 . x7) (λ x9 . x0 (λ x10 : (ι → ι) → ι . 0) (λ x10 : ι → ι . Inj1 0) (λ x10 : (ι → ι) → ι . setsum 0 0)) (setsum 0 0) (x1 (λ x9 . x7) (λ x9 . x0 (λ x10 : (ι → ι) → ι . 0) (λ x10 : ι → ι . 0) (λ x10 : (ι → ι) → ι . 0)) (x3 (λ x9 : (((ι → ι) → ι → ι) → ι) → ((ι → ι) → ι → ι) → ι . 0) (λ x9 x10 x11 x12 . 0) 0) (Inj0 0) (λ x9 . 0)) (λ x9 . x9))) ... ... ... = ....