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