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 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 x5 x6 x7 . x3 (λ x8 x9 . x8) (λ x8 . x1 (λ x9 . 0) x5) = Inj0 (x3 (λ x8 x9 . 0) (λ x8 . x7)).
Assume H1:
∀ x4 . ∀ x5 : ι → (ι → ι) → (ι → ι) → ι → ι . ∀ x6 . ∀ x7 : (((ι → ι) → ι) → ι) → ι . x3 (λ x8 x9 . 0) (λ x8 . x5 x8 (λ x9 . x7 (λ x10 : (ι → ι) → ι . 0)) (λ x9 . setsum 0 (x1 (λ x10 . x9) 0)) (x5 (setsum (setsum 0 0) (x5 0 (λ x9 . 0) (λ x9 . 0) 0)) (λ x9 . setsum x6 (setsum 0 0)) (λ x9 . Inj0 (x3 (λ x10 x11 . 0) (λ x10 . 0))) (setsum x8 0))) = x5 (x3 (λ x8 x9 . 0) (λ x8 . Inj0 0)) (λ x8 . x1 (λ x9 . x2 (λ x10 . 0) (setsum (x7 (λ x10 : (ι → ι) → ι . 0)) (setsum 0 0)) (λ x10 : ι → ι → ι . λ x11 : ι → ι . x0 (λ x12 : ι → ι . λ x13 . λ x14 : ι → ι → ι . λ x15 x16 . 0) (Inj1 0) (x2 (λ x12 . 0) 0 (λ x12 : ι → ι → ι . λ x13 : ι → ι . 0) (λ x12 : ι → ι . 0))) (λ x10 : ι → ι . 0)) 0) (λ x8 . x7 (λ x9 : (ι → ι) → ι . 0)) (Inj1 0).
Assume H2:
∀ x4 x5 . ∀ x6 : ι → ι . ∀ x7 . x2 (λ x8 . x6 0) (x0 (λ x8 : ι → ι . λ x9 . λ x10 : ι → ι → ι . λ x11 x12 . x1 (λ x13 . Inj1 x13) (setsum (setsum 0 0) (Inj0 0))) x4 (setsum (Inj1 (Inj0 0)) (Inj0 x4))) (λ x8 : ι → ι → ι . λ x9 : ι → ι . 0) (λ x8 : ι → ι . 0) = setsum 0 (setsum (setsum 0 (setsum 0 (setsum 0 0))) (x0 (λ x8 : ι → ι . λ x9 . λ x10 : ι → ι → ι . λ x11 x12 . setsum x12 x12) x7 (x1 (λ x8 . x5) 0))).
Assume H3:
∀ x4 x5 : ι → ι . ∀ x6 : ι → ι → ι . ∀ x7 . x2 (λ x8 . setsum (x5 (x3 (λ x9 x10 . x10) (λ x9 . Inj0 0))) (setsum (x0 (λ x9 : ι → ι . λ x10 . λ x11 : ι → ι → ι . λ x12 x13 . x10) 0 (x3 (λ x9 x10 . 0) (λ x9 . 0))) (x3 (λ x9 x10 . 0) (λ x9 . 0)))) (x1 (λ x8 . 0) x7) (λ x8 : ι → ι → ι . λ x9 : ι → ι . Inj1 (x1 (λ x10 . 0) (x3 (λ x10 x11 . Inj0 0) (λ x10 . x2 (λ x11 . 0) 0 (λ x11 : ι → ι → ι . λ x12 : ι → ι . 0) (λ x11 : ι → ι . 0))))) (λ x8 : ι → ι . x0 (λ x9 : ι → ι . λ x10 . λ x11 : ι → ι → ι . λ x12 x13 . x12) 0 0) = Inj1 x7.
Assume H4:
∀ x4 x5 x6 x7 . x1 (λ x8 . x1 (λ x9 . 0) (setsum 0 x5)) ... = ....