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