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