Let x0 of type (ι → ι) → ι → ι be given.
Let x1 of type (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 be given.
Let x3 of type (((ι → ι) → ι) → ι) → ι → ι be given.
Assume H0:
∀ x4 : ι → ι . ∀ x5 : ι → ι → ι . ∀ x6 : ι → (ι → ι) → ι → ι . ∀ x7 . x3 (λ x8 : (ι → ι) → ι . Inj0 0) (x6 (setsum 0 0) (λ x8 . 0) (x4 0)) = x6 (Inj0 (Inj0 (Inj1 (Inj1 0)))) (λ x8 . setsum (x2 (λ x9 . λ x10 : ((ι → ι) → ι → ι) → ι . λ x11 : ι → ι . λ x12 x13 . x3 (λ x14 : (ι → ι) → ι . x2 (λ x15 . λ x16 : ((ι → ι) → ι → ι) → ι . λ x17 : ι → ι . λ x18 x19 . 0) 0) (x2 (λ x14 . λ x15 : ((ι → ι) → ι → ι) → ι . λ x16 : ι → ι . λ x17 x18 . 0) 0)) (setsum 0 0)) x8) x7.
Assume H1:
∀ x4 : ι → (ι → ι → ι) → (ι → ι) → ι → ι . ∀ x5 : ι → (ι → ι) → ι → ι . ∀ x6 x7 . x3 (λ x8 : (ι → ι) → ι . Inj0 0) (x3 (λ x8 : (ι → ι) → ι . 0) (x1 (λ x8 . λ x9 : (ι → ι) → ι . x6) x7)) = x3 (λ x8 : (ι → ι) → ι . x5 0 (λ x9 . x1 (λ x10 . λ x11 : (ι → ι) → ι . x3 (λ x12 : (ι → ι) → ι . 0) (x1 (λ x12 . λ x13 : (ι → ι) → ι . 0) 0)) (Inj0 0)) 0) (Inj0 (x3 (λ x8 : (ι → ι) → ι . x0 (λ x9 . x8 (λ x10 . 0)) x7) x6)).
Assume H2:
∀ x4 x5 . ∀ x6 : ι → ι → ι . ∀ x7 : ((ι → ι → ι) → (ι → ι) → ι → ι) → ι . x2 (λ x8 . λ x9 : ((ι → ι) → ι → ι) → ι . λ x10 : ι → ι . λ x11 x12 . Inj0 (x2 (λ x13 . λ x14 : ((ι → ι) → ι → ι) → ι . λ x15 : ι → ι . λ x16 x17 . 0) 0)) (setsum (setsum (Inj1 0) (Inj1 (x1 (λ x8 . λ x9 : (ι → ι) → ι . 0) 0))) 0) = Inj0 0.
Assume H3:
∀ x4 : ι → (ι → ι) → ι → ι → ι . ∀ x5 . ∀ x6 : (ι → (ι → ι) → ι) → ι → ι . ∀ x7 . x2 (λ x8 . λ x9 : ((ι → ι) → ι → ι) → ι . λ x10 : ι → ι . λ x11 x12 . x1 (λ x13 . λ x14 : (ι → ι) → ι . x3 (λ x15 : (ι → ι) → ι . x3 (λ x16 : (ι → ι) → ι . 0) (Inj0 0)) (setsum 0 0)) (Inj0 x12)) (x0 (λ x8 . Inj1 (setsum (x3 (λ x9 : (ι → ι) → ι . 0) 0) (x3 (λ x9 : (ι → ι) → ι . 0) 0))) (setsum (setsum (x1 (λ x8 . λ x9 : (ι → ι) → ι . 0) 0) 0) (Inj1 (x3 (λ x8 : (ι → ι) → ι . 0) 0)))) = setsum (setsum (x2 (λ x8 . λ x9 : ((ι → ι) → ι → ι) → ι . λ x10 : ι → ι . λ x11 x12 . x12) (x2 (λ x8 . λ x9 : ((ι → ι) → ι → ι) → ι . λ x10 : ι → ι . λ x11 x12 . setsum 0 0) x7)) 0) (setsum (x1 (λ x8 . λ x9 : (ι → ι) → ι . 0) x7) (x3 (λ x8 : (ι → ι) → ι . x0 (λ x9 . x7) (x1 (λ x9 . λ x10 : (ι → ι) → ι . 0) 0)) (x4 (x3 (λ x8 : (ι → ι) → ι . 0) 0) (λ x8 . setsum 0 0) (x6 (λ x8 . λ x9 : ι → ι . 0) 0) (x6 (λ x8 . λ x9 : ι → ι . 0) 0)))).
Assume H4:
∀ x4 : (ι → (ι → ι) → ι) → ι → ι . ∀ x5 . ∀ x6 : ι → ι → ι → ι . ∀ x7 : ι → ι . x1 (λ x8 . λ x9 : (ι → ι) → ι . x1 (λ x10 . λ x11 : (ι → ι) → ι . setsum x10 0) (x9 (λ x10 . 0))) (x4 (λ x8 . λ x9 : ι → ι . setsum (Inj1 ...) ...) 0) = ....