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 be given.
Let x2 of type (ι → ι) → ι → ι be given.
Let x3 of type ((ι → ι) → (CN (ι → ι)) → ι) → ι → ι be given.
Assume H0:
∀ x4 : ((ι → ι → ι) → ι) → (ι → ι → ι) → ι → ι . ∀ x5 : (ι → ι) → ((ι → ι) → ι) → ι . ∀ x6 : ι → (ι → ι) → ι → ι . ∀ x7 . x3 (λ x8 : ι → ι . λ x9 : ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . x6 (x9 (λ x10 : ι → ι . λ x11 . setsum (setsum 0 0) (Inj0 0)) (λ x10 . x0 (λ x11 . 0) (λ x11 : ((ι → ι) → ι) → ι . λ x12 x13 . setsum 0 0)) 0) (λ x10 . setsum (x1 (λ x11 : ((ι → ι) → ι) → ι . x0 (λ x12 . 0) (λ x12 : ((ι → ι) → ι) → ι . λ x13 x14 . 0)) (λ x11 . 0)) 0) (x2 (λ x10 . setsum 0 (x0 (λ x11 . 0) (λ x11 : ((ι → ι) → ι) → ι . λ x12 x13 . 0))) 0)) (x4 (λ x8 : ι → ι → ι . x1 (λ x9 : ((ι → ι) → ι) → ι . x9 (λ x10 : ι → ι . x1 (λ x11 : ((ι → ι) → ι) → ι . 0) (λ x11 . 0))) (λ x9 . x0 (λ x10 . setsum 0 0) (λ x10 : ((ι → ι) → ι) → ι . λ x11 x12 . setsum 0 0))) (λ x8 x9 . x7) (setsum 0 (x6 x7 (λ x8 . Inj1 0) 0))) = setsum x7 (x4 (λ x8 : ι → ι → ι . setsum (x1 (λ x9 : ((ι → ι) → ι) → ι . Inj1 0) (λ x9 . setsum 0 0)) (x3 (λ x9 : ι → ι . λ x10 : ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . Inj1 0) (setsum 0 0))) (λ x8 x9 . x1 (λ x10 : ((ι → ι) → ι) → ι . x3 (λ x11 : ι → ι . λ x12 : ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . setsum 0 0) (Inj0 0)) (λ x10 . x7)) (x3 (λ x8 : ι → ι . λ x9 : ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . setsum (x6 0 (λ x10 . 0) 0) (setsum 0 0)) (x4 (λ x8 : ι → ι → ι . setsum 0 0) (λ x8 x9 . x8) (Inj1 0)))).
Assume H1:
∀ x4 . ∀ x5 : (ι → (ι → ι) → ι → ι) → (ι → ι) → (ι → ι) → ι → ι . ∀ x6 x7 . x3 (λ x8 : ι → ι . λ x9 : ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . x3 (λ x10 : ι → ι . λ x11 : ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . Inj0 0) (x9 (λ x10 : ι → ι . λ x11 . 0) (λ x10 . x0 (λ x11 . x1 (λ x12 : ((ι → ι) → ι) → ι . 0) (λ x12 . 0)) (λ x11 : ((ι → ι) → ι) → ι . λ x12 x13 . Inj1 0)) 0)) (x3 (λ x8 : ι → ι . λ x9 : ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . 0) (x5 (λ x8 . λ x9 : ι → ι . λ x10 . x7) (λ x8 . setsum (x2 (λ x9 . 0) 0) (x1 (λ x9 : ((ι → ι) → ι) → ι . 0) (λ x9 . 0))) (λ x8 . Inj0 x6) (setsum (setsum 0 0) (setsum 0 0)))) = Inj1 (setsum 0 0).
Assume H2:
∀ x4 . ∀ x5 : (ι → ι) → ι . ∀ x6 : (((ι → ι) → ι → ι) → ι) → ι . ∀ x7 : ι → ι . x2 (λ x8 . 0) (setsum (x6 (λ x8 : (ι → ι) → ι → ι . x3 (λ x9 : ι → ι . λ x10 : ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . x1 (λ x11 : ((ι → ι) → ι) → ι . 0) (λ x11 . 0)) (Inj0 0))) (setsum 0 (Inj0 (x2 (λ x8 . 0) 0)))) = x6 (λ x8 : (ι → ι) → ι → ι . x0 (λ x9 . x7 0) (λ x9 : ((ι → ι) → ι) → ι . λ x10 x11 . setsum 0 (x8 (λ x12 . setsum 0 0) 0))).
Assume H3: ∀ x4 : (((ι → ι) → ι) → ι) → ι . ∀ x5 . ∀ x6 : ι → ι . ....