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