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