Let x0 of type (ι → ι → ι → ι) → ι → ι → ι → ι be given.
Let x1 of type (ι → (ι → ι) → ι) → CT4 ι be given.
Let x2 of type (ι → (ι → ι) → ((ι → ι) → ι) → (ι → ι) → ι → ι) → (ι → ι) → ι be given.
Let x3 of type (ι → ι) → ι → ι → ι be given.
Assume H0:
∀ x4 x5 x6 x7 . x3 (λ x8 . x3 (λ x9 . x7) (x3 (λ x9 . x7) 0 (Inj0 0)) x5) 0 (setsum (Inj0 (setsum (x0 (λ x8 x9 x10 . 0) 0 0 0) x5)) (x1 (λ x8 . λ x9 : ι → ι . x2 (λ x10 . λ x11 : ι → ι . λ x12 : (ι → ι) → ι . λ x13 : ι → ι . λ x14 . setsum 0 0) (λ x10 . x10)) (λ x8 x9 x10 x11 . x2 (λ x12 . λ x13 : ι → ι . λ x14 : (ι → ι) → ι . λ x15 : ι → ι . λ x16 . x14 (λ x17 . 0)) (λ x12 . x12)))) = setsum (x0 (λ x8 x9 x10 . 0) x6 0 (setsum 0 (Inj0 (Inj1 0)))) x7.
Assume H1:
∀ x4 : ((ι → ι) → (ι → ι) → ι → ι) → ι . ∀ x5 : (((ι → ι) → ι → ι) → ι → ι) → ι → (ι → ι) → ι . ∀ x6 : (((ι → ι) → ι) → (ι → ι) → ι) → ι . ∀ x7 . x3 (λ x8 . x6 (λ x9 : (ι → ι) → ι . λ x10 : ι → ι . x2 (λ x11 . λ x12 : ι → ι . λ x13 : (ι → ι) → ι . λ x14 : ι → ι . λ x15 . 0) (λ x11 . x9 (λ x12 . x1 (λ x13 . λ x14 : ι → ι . 0) (λ x13 x14 x15 x16 . 0))))) (setsum (setsum 0 0) (x5 (λ x8 : (ι → ι) → ι → ι . λ x9 . x2 (λ x10 . λ x11 : ι → ι . λ x12 : (ι → ι) → ι . λ x13 : ι → ι . λ x14 . 0) (λ x10 . setsum 0 0)) (x3 (λ x8 . x8) x7 (Inj1 0)) (λ x8 . 0))) (x3 (λ x8 . x6 (λ x9 : (ι → ι) → ι . λ x10 : ι → ι . x3 (λ x11 . Inj0 0) (x10 0) (x2 (λ x11 . λ x12 : ι → ι . λ x13 : (ι → ι) → ι . λ x14 : ι → ι . λ x15 . 0) (λ x11 . 0)))) (x6 (λ x8 : (ι → ι) → ι . λ x9 : ι → ι . setsum 0 (x0 (λ x10 x11 x12 . 0) 0 0 0))) (setsum (x5 (λ x8 : (ι → ι) → ι → ι . λ x9 . x8 (λ x10 . 0) 0) (setsum 0 0) (λ x8 . x1 (λ x9 . λ x10 : ι → ι . 0) (λ x9 x10 x11 x12 . 0))) (Inj0 0))) = Inj0 (x2 (λ x8 . λ x9 : ι → ι . λ x10 : (ι → ι) → ι . λ x11 : ι → ι . λ x12 . x2 (λ x13 . λ x14 : ι → ι . λ x15 : (ι → ι) → ι . λ x16 : ι → ι . λ x17 . 0) (λ x13 . x10 (λ x14 . x11 0))) (λ x8 . 0)).
Assume H2:
∀ x4 : ι → ι . ∀ x5 x6 x7 . x2 (λ x8 . λ x9 : ι → ι . λ x10 : (ι → ι) → ι . λ x11 : ι → ι . λ x12 . Inj1 (Inj1 0)) (λ x8 . x5) = Inj0 (x3 (λ x8 . Inj1 x8) (setsum (setsum (x1 (λ x8 . λ x9 : ι → ι . 0) (λ x8 x9 x10 x11 . 0)) (x2 (λ x8 . λ x9 : ι → ι . λ x10 : (ι → ι) → ι . λ x11 : ι → ι . λ x12 . 0) (λ x8 . 0))) (x2 (λ x8 . λ x9 : ι → ι . λ x10 : (ι → ι) → ι . λ x11 : ι → ι . λ x12 . x12) (λ x8 . x1 (λ x9 . λ x10 : ι → ι . 0) (λ x9 x10 x11 x12 . 0)))) (x3 (λ x8 . x2 (λ x9 . λ x10 : ι → ι . λ x11 : (ι → ι) → ι . λ x12 : ι → ι . λ x13 . 0) (λ x9 . x2 (λ x10 . λ x11 : ι → ι . λ x12 : (ι → ι) → ι . λ x13 : ι → ι . ...) ...)) ... ...)).