Let x0 of type (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 be given.
Let x3 of type (ι → ι) → (ι → ι → ι) → ι → ((ι → ι) → ι) → ι be given.
Assume H0:
∀ x4 . ∀ x5 : ι → (ι → ι → ι) → ι . ∀ x6 : (ι → ι) → (ι → ι → ι) → ι . ∀ x7 : (ι → ι) → ι . x3 (λ x8 . x0 (λ x9 . Inj1 x8) (λ x9 x10 . λ x11 : ι → ι . x11 (x2 (λ x12 . setsum 0 0) (Inj1 0)))) (λ x8 x9 . x2 (λ x10 . 0) (setsum (x6 (λ x10 . x1 (λ x11 : ((ι → ι) → (ι → ι) → ι) → ι . λ x12 x13 . λ x14 : ι → ι . 0) 0) (λ x10 x11 . 0)) (x3 (λ x10 . 0) (λ x10 x11 . setsum 0 0) (setsum 0 0) (λ x10 : ι → ι . x10 0)))) (Inj0 (setsum (setsum (x6 (λ x8 . 0) (λ x8 x9 . 0)) 0) (x0 (λ x8 . setsum 0 0) (λ x8 x9 . λ x10 : ι → ι . x9)))) (λ x8 : ι → ι . 0) = setsum 0 0.
Assume H1:
∀ x4 : (ι → ι → ι) → ι → (ι → ι) → ι . ∀ x5 : ι → ι . ∀ x6 . ∀ x7 : ι → ι . x3 (λ x8 . x8) (λ x8 x9 . Inj1 0) 0 (λ x8 : ι → ι . setsum (Inj0 (Inj1 0)) (Inj1 (setsum (Inj0 0) (x8 0)))) = setsum 0 (setsum (x2 (λ x8 . x2 (λ x9 . setsum 0 0) (Inj1 0)) (x5 (x3 (λ x8 . 0) (λ x8 x9 . 0) 0 (λ x8 : ι → ι . 0)))) (x0 (λ x8 . x8) (λ x8 x9 . λ x10 : ι → ι . x2 (λ x11 . 0) (x1 (λ x11 : ((ι → ι) → (ι → ι) → ι) → ι . λ x12 x13 . λ x14 : ι → ι . 0) 0)))).
Assume H2:
∀ x4 x5 . ∀ x6 : (((ι → ι) → ι) → (ι → ι) → ι → ι) → (ι → ι → ι) → ι . ∀ x7 . x2 (λ x8 . 0) 0 = setsum (setsum x4 (setsum x7 x7)) (Inj0 (x0 (λ x8 . 0) (λ x8 x9 . λ x10 : ι → ι . x8))).
Assume H3:
∀ x4 : ι → ι . ∀ x5 x6 x7 . x2 (λ x8 . x1 (λ x9 : ((ι → ι) → (ι → ι) → ι) → ι . λ x10 x11 . λ x12 : ι → ι . Inj1 0) x8) 0 = x1 (λ x8 : ((ι → ι) → (ι → ι) → ι) → ι . λ x9 x10 . λ x11 : ι → ι . x2 (λ x12 . 0) (x1 (λ x12 : ((ι → ι) → (ι → ι) → ι) → ι . λ x13 x14 . λ x15 : ι → ι . Inj1 (x0 (λ x16 . 0) (λ x16 x17 . λ x18 : ι → ι . 0))) (setsum 0 x9))) (x4 (Inj1 (Inj0 0))).
Assume H4:
∀ x4 : (ι → ι) → ι . ∀ x5 x6 x7 . x1 (λ x8 : ((ι → ι) → (ι → ι) → ι) → ι . λ x9 x10 . λ x11 : ι → ι . setsum (setsum 0 (x2 (λ x12 . 0) (x11 0))) (x1 (λ x12 : ((ι → ι) → (ι → ι) → ι) → ι . λ x13 x14 . λ x15 : ι → ι . Inj1 (x0 (λ x16 . 0) (λ x16 x17 . λ x18 : ι → ι . 0))) 0)) (x2 (λ x8 . setsum (Inj1 (x2 (λ x9 . 0) 0)) 0) 0) = setsum (Inj0 0) 0.
Assume H5:
∀ x4 . ∀ x5 : (((ι → ι) → ι) → ι → ι) → ((ι → ι) → ι) → ι . ∀ x6 : ((ι → ι → ι) → ι) → ((ι → ι) → ι) → ι . ∀ x7 : ι → ι . x1 (λ x8 : ((ι → ι) → (ι → ι) → ι) → ι . λ x9 x10 . λ x11 : ι → ι . 0) (setsum (Inj0 (x6 (λ x8 : ι → ι → ι . x2 (λ x9 . 0) 0) (λ x8 : ι → ι . x1 (λ x9 : ((ι → ι) → (ι → ι) → ι) → ι . λ x10 x11 . λ x12 : ι → ι . 0) 0))) ...) = ....