Let x0 of type ((ι → ι → ι → ι) → ((ι → ι → ι) → ι → ι → ι) → ι) → ι → ο be given.
Let x1 of type (ι → ι → ι) → ι → ι → ο be given.
Let x2 of type (ι → ι) → ((ι → (ι → ι) → ι) → ι → ι) → ο be given.
Let x3 of type (ι → ι → ι) → ι → ο be given.
Assume H0:
∀ x4 : ι → ι → ι → ι → ι . ∀ x5 x6 . ∀ x7 : ι → ι . x0 (λ x8 : ι → ι → ι → ι . λ x9 : (ι → ι → ι) → ι → ι → ι . setsum (Inj1 (Inj1 (Inj0 0))) x6) (Inj0 (x7 (x7 (Inj1 0)))) ⟶ x3 (λ x8 x9 . 0) (setsum 0 (setsum x5 (Inj1 (x4 0 0 0 0)))).
Assume H1:
∀ x4 x5 x6 . ∀ x7 : (ι → ι) → (ι → ι → ι) → ι . In (Inj0 x5) x4 ⟶ x3 (λ x8 x9 . 0) (setsum (setsum (Inj1 x4) x5) (setsum (x7 (λ x8 . x6) (λ x8 x9 . 0)) (x7 (λ x8 . 0) (λ x8 x9 . 0)))) ⟶ x1 (λ x8 x9 . setsum (Inj1 (x7 (λ x10 . Inj0 0) (λ x10 x11 . 0))) (Inj1 0)) 0 0.
Assume H2:
∀ x4 . ∀ x5 : ((ι → ι → ι) → ι → ι) → ι . ∀ x6 x7 . In (Inj1 (Inj0 0)) (Inj0 (Inj0 x7)) ⟶ x0 (λ x8 : ι → ι → ι → ι . λ x9 : (ι → ι → ι) → ι → ι → ι . 0) (setsum (x5 (λ x8 : ι → ι → ι . λ x9 . setsum (setsum 0 0) (x8 0 0))) x7) ⟶ x2 (λ x8 . Inj1 (Inj0 (setsum (setsum 0 0) x8))) (λ x8 : ι → (ι → ι) → ι . λ x9 . 0).
Assume H3:
∀ x4 . ∀ x5 : ι → (ι → ι) → ι . ∀ x6 : ι → ι → ι . ∀ x7 . x2 (λ x8 . 0) (λ x8 : ι → (ι → ι) → ι . λ x9 . Inj0 (Inj1 x7)) ⟶ False.
Assume H6:
∀ x4 : ι → ι → ι . ∀ x5 x6 x7 . In (Inj0 (setsum (setsum (setsum 0 0) (setsum 0 0)) x5)) (Inj1 x6) ⟶ x0 (λ x8 : ι → ι → ι → ι . λ x9 : (ι → ι → ι) → ι → ι → ι . 0) x5.
Assume H7:
∀ x4 : ι → ι . ∀ x5 x6 x7 . x0 (λ x8 : ι → ι → ι → ι . λ x9 : (ι → ι → ι) → ι → ι → ι . 0) 0 ⟶ False.
Apply unknownprop_92e9d75f0c2736874e0d273c2aebd9f3628211a178962928cb4fc08e22c09f27 with
λ x4 x5 . In (Inj0 (setsum ... 0)) ....
Apply L8.
Apply L9 with
0.
The subproof is completed by applying L10.