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