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