Let x0 of type (ι → ι) → ι → ι → ((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 → 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 → ι be given.
Let x3 of type ((ι → ι) → (((ι → ι) → ι → ι) → ι → ι) → ι) → ι → ι → ι be given.
Assume H0:
∀ x4 . ∀ x5 : ι → ι → ι . ∀ x6 : ι → ι . ∀ x7 : (ι → ι) → ι . x3 (λ x8 : ι → ι . λ x9 : ((ι → ι) → ι → ι) → ι → ι . 0) (Inj1 (x2 (λ x8 : ((ι → ι → ι) → (ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ι . λ x9 . λ x10 : (ι → ι) → ι → ι . x0 (λ x11 . x11) (setsum 0 0) (Inj0 0) (λ x11 : ι → ι . λ x12 . setsum 0 0)) (x3 (λ x8 : ι → ι . λ x9 : ((ι → ι) → ι → ι) → ι → ι . x0 (λ x10 . 0) 0 0 (λ x10 : ι → ι . λ x11 . 0)) (x6 0) (setsum 0 0)))) (x6 0) = x6 (Inj1 0).
Assume H1:
∀ x4 . ∀ x5 : ι → ι → ι → ι . ∀ x6 x7 . x3 (λ x8 : ι → ι . λ x9 : ((ι → ι) → ι → ι) → ι → ι . setsum 0 (Inj0 0)) (x2 (λ x8 : ((ι → ι → ι) → (ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ι . λ x9 . λ x10 : (ι → ι) → ι → ι . 0) (x3 (λ x8 : ι → ι . λ x9 : ((ι → ι) → ι → ι) → ι → ι . Inj1 (setsum 0 0)) (setsum (Inj0 0) x4) 0)) (x2 (λ x8 : ((ι → ι → ι) → (ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ι . λ x9 . λ x10 : (ι → ι) → ι → ι . 0) 0) = x2 (λ x8 : ((ι → ι → ι) → (ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ι . λ x9 . λ x10 : (ι → ι) → ι → ι . setsum (x3 (λ x11 : ι → ι . λ x12 : ((ι → ι) → ι → ι) → ι → ι . x11 (x12 (λ x13 : ι → ι . λ x14 . 0) 0)) (setsum (x3 (λ x11 : ι → ι . λ x12 : ((ι → ι) → ι → ι) → ι → ι . 0) 0 0) x7) x7) (Inj1 (x1 (λ x11 : ι → ι . λ x12 : (ι → ι) → (ι → ι) → ι . λ x13 x14 x15 . x14) (λ x11 : ((ι → ι) → ι) → ι → ι → ι . λ x12 : (ι → ι) → ι . λ x13 : ι → ι . 0)))) (setsum (x5 0 (x3 (λ x8 : ι → ι . λ x9 : ((ι → ι) → ι → ι) → ι → ι . Inj1 0) x4 (x3 (λ x8 : ι → ι . λ x9 : ((ι → ι) → ι → ι) → ι → ι . 0) 0 0)) (setsum (Inj0 0) (x3 (λ x8 : ι → ι . λ x9 : ((ι → ι) → ι → ι) → ι → ι . 0) 0 0))) 0).
Assume H2:
∀ x4 x5 . ∀ x6 : ι → ι . ∀ x7 . x2 (λ x8 : ((ι → ι → ι) → (ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ι . λ x9 . λ x10 : (ι → ι) → ι → ι . x1 (λ x11 : ι → ι . λ x12 : (ι → ι) → (ι → ι) → ι . λ x13 x14 x15 . setsum 0 (setsum 0 0)) (λ x11 : ((ι → ι) → ι) → ι → ι → ι . λ x12 : (ι → ι) → ι . λ x13 : ι → ι . 0)) (Inj0 (x1 (λ x8 : ι → ι . λ x9 : (ι → ι) → (ι → ι) → ι . λ x10 x11 x12 . setsum (setsum 0 0) (x3 (λ x13 : ι → ι . λ x14 : ((ι → ι) → ι → ι) → ι → ι . 0) 0 0)) (λ x8 : ((ι → ι) → ι) → ι → ι → ι . λ x9 : (ι → ι) → ι . λ x10 : ι → ι . 0))) = x1 (λ x8 : ι → ι . λ x9 : (ι → ι) → (ι → ι) → ι . λ x10 x11 x12 . setsum (Inj1 (setsum 0 (Inj1 0))) (Inj0 (Inj0 0))) (λ x8 : ((ι → ι) → ι) → ι → ι → ι . λ x9 : (ι → ι) → ι . λ x10 : ι → ι . Inj0 (x9 (λ x11 . setsum (x1 (λ x12 : ι → ι . λ x13 : (ι → ι) → (ι → ι) → ι . λ x14 x15 x16 . 0) (λ x12 : ((ι → ι) → ι) → ι → ι → ι . λ x13 : (ι → ι) → ι . λ x14 : ι → ι . 0)) (x3 (λ x12 : ι → ι . λ x13 : ((ι → ι) → ι → ι) → ι → ι . 0) 0 0)))).
Assume H3:
∀ x4 : ι → ((ι → ι) → ι → ι) → ι . ∀ x5 : (((ι → ι) → ι → ι) → (ι → ι) → ι) → ι → ι . ∀ x6 . ∀ x7 : (((ι → ι) → ι → ι) → (ι → ι) → ι → ι) → ι → ι . x2 (λ x8 : ((ι → ι → ι) → (ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ι . λ x9 . λ x10 : (ι → ι) → ι → ι . x8 (λ x11 : ι → ι → ι . λ x12 : ι → ι . λ x13 . setsum (x1 (λ x14 : ι → ι . λ x15 : (ι → ι) → (ι → ι) → ι . λ x16 x17 x18 . 0) (λ x14 : ((ι → ι) → ι) → ι → ι → ι . λ x15 : (ι → ι) → ι . λ x16 : ι → ι . Inj0 0)) (Inj1 (setsum 0 0))) (λ x11 : ι → ι . λ x12 . 0)) (x5 (λ x8 : (ι → ι) → ι → ι . λ x9 : ι → ι . x9 (x0 ... ... ... ...)) ...) = ....