Let x0 of type (ι → ι) → CT2 ι be given.
Let x1 of type (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 be given.
Assume H0:
∀ x4 : (((ι → ι) → ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . ∀ x5 x6 . ∀ x7 : ((ι → ι) → (ι → ι) → ι → ι) → ι → (ι → ι) → ι . x3 (λ x8 : (ι → ι) → ι . 0) (setsum 0 (setsum (x4 (λ x8 : (ι → ι) → ι → ι . λ x9 . x2 (λ x10 . 0) 0 (λ x10 . 0) 0 (λ x10 . 0)) (λ x8 : ι → ι . λ x9 . x8 0) (λ x8 . Inj1 0) 0) (Inj0 (Inj1 0)))) (x4 (λ x8 : (ι → ι) → ι → ι . λ x9 . x1 (λ x10 . setsum 0 0) (λ x10 x11 . λ x12 : ι → ι . x1 (λ x13 . x10) (λ x13 x14 . λ x15 : ι → ι . x14))) (λ x8 : ι → ι . λ x9 . x6) (λ x8 . 0) (x4 (λ x8 : (ι → ι) → ι → ι . λ x9 . x1 (λ x10 . Inj0 0) (λ x10 x11 . λ x12 : ι → ι . x2 (λ x13 . 0) 0 (λ x13 . 0) 0 (λ x13 . 0))) (λ x8 : ι → ι . λ x9 . setsum 0 (setsum 0 0)) (λ x8 . 0) (Inj1 (setsum 0 0)))) = Inj0 x5.
Assume H1:
∀ x4 . ∀ x5 : (((ι → ι) → ι → ι) → ι) → ι . ∀ x6 : ι → ι . ∀ x7 . x3 (λ x8 : (ι → ι) → ι . x0 (λ x9 . Inj1 0) (λ x9 x10 . x0 (λ x11 . x8 (λ x12 . x11)) (λ x11 x12 . x2 (λ x13 . 0) (Inj0 0) (λ x13 . 0) (setsum 0 0) (λ x13 . x11)))) (setsum (Inj0 (x0 (λ x8 . x2 (λ x9 . 0) 0 (λ x9 . 0) 0 (λ x9 . 0)) (λ x8 x9 . x7))) 0) (Inj1 0) = setsum (setsum x7 0) 0.
Assume H2:
∀ x4 x5 . ∀ x6 : ι → ι → (ι → ι) → ι → ι . ∀ x7 . x2 (λ x8 . setsum 0 x8) x4 (λ x8 . x2 (λ x9 . x9) (x2 (λ x9 . x2 (λ x10 . 0) 0 (λ x10 . 0) (x0 (λ x10 . 0) (λ x10 x11 . 0)) (λ x10 . Inj1 0)) x8 (λ x9 . x2 (λ x10 . x8) (x1 (λ x10 . 0) (λ x10 x11 . λ x12 : ι → ι . 0)) (λ x10 . x10) (x2 (λ x10 . 0) 0 (λ x10 . 0) 0 (λ x10 . 0)) (λ x10 . 0)) (x1 (λ x9 . x9) (λ x9 x10 . λ x11 : ι → ι . x2 (λ x12 . 0) 0 (λ x12 . 0) 0 (λ x12 . 0))) (λ x9 . x8)) (λ x9 . x8) (setsum (setsum (Inj1 0) (Inj1 0)) 0) (λ x9 . x3 (λ x10 : (ι → ι) → ι . x3 (λ x11 : (ι → ι) → ι . x1 (λ x12 . 0) (λ x12 x13 . λ x14 : ι → ι . 0)) (x1 (λ x11 . 0) (λ x11 x12 . λ x13 : ι → ι . 0)) (setsum 0 0)) x7 (x1 (λ x10 . x0 (λ x11 . 0) (λ x11 x12 . 0)) (λ x10 x11 . λ x12 : ι → ι . 0)))) ... ... = ....