Let x0 of type ((ι → ι) → (((ι → ι) → ι) → ι) → ι → (ι → ι) → ι → ι) → ι → ι be given.
Let x1 of type (ι → ι) → ι → ι be given.
Let x2 of type (ι → ι) → ι → ι be given.
Let x3 of type (ι → ι) → (ι → ι) → ι be given.
Assume H0:
∀ x4 x5 . ∀ x6 : (((ι → ι) → ι → ι) → (ι → ι) → ι) → ι . ∀ x7 . x3 (λ x8 . x2 (λ x9 . x7) 0) (λ x8 . x7) = x2 (λ x8 . x5) (setsum (setsum (Inj0 0) 0) x4).
Assume H1:
∀ x4 : (((ι → ι) → ι → ι) → ι → ι) → ι → ι . ∀ x5 : (ι → ι) → ι → ι . ∀ x6 : ι → ι . ∀ x7 : (ι → (ι → ι) → ι) → ι . x3 (λ x8 . x0 (λ x9 : ι → ι . λ x10 : ((ι → ι) → ι) → ι . λ x11 . λ x12 : ι → ι . λ x13 . x3 (λ x14 . x14) (λ x14 . Inj1 (x1 (λ x15 . 0) 0))) 0) (λ x8 . x7 (λ x9 . λ x10 : ι → ι . 0)) = x7 (λ x8 . λ x9 : ι → ι . Inj0 (x6 x8)).
Assume H2:
∀ x4 . ∀ x5 : ι → ι . ∀ x6 . ∀ x7 : ι → (ι → ι) → ι . x2 (λ x8 . x8) (setsum x6 (x0 (λ x8 : ι → ι . λ x9 : ((ι → ι) → ι) → ι . λ x10 . λ x11 : ι → ι . λ x12 . setsum 0 0) 0)) = x5 x4.
Apply FalseE with
(∀ x4 : ι → (ι → ι) → ι . ∀ x5 x6 . ∀ x7 : (ι → ι → ι → ι) → ι → ι . x2 (λ x8 . 0) (Inj1 (x2 (λ x8 . x7 (λ x9 x10 x11 . setsum 0 0) (Inj1 0)) (x2 (λ x8 . x8) (Inj0 0)))) = x7 (λ x8 x9 x10 . x0 (λ x11 : ι → ι . λ x12 : ((ι → ι) → ι) → ι . λ x13 . λ x14 : ι → ι . λ x15 . Inj1 (Inj0 x15)) x10) 0) ⟶ (∀ x4 x5 x6 . ∀ x7 : (ι → ι → ι → ι) → ι → ι . x1 (λ x8 . x5) 0 = Inj1 x6) ⟶ (∀ x4 : ι → ι → ι . ∀ x5 x6 . ∀ x7 : ι → ((ι → ι) → ι → ι) → ι . x1 (λ x8 . x3 (λ x9 . 0) (λ x9 . x8)) (x0 (λ x8 : ι → ι . λ x9 : ((ι → ι) → ι) → ι . λ x10 . λ x11 : ι → ι . λ x12 . 0) (x4 (setsum (x4 0 0) 0) (Inj1 x6))) = x0 (λ x8 : ι → ι . λ x9 : ((ι → ι) → ι) → ι . λ x10 . λ x11 : ι → ι . λ x12 . x11 (x3 (λ x13 . x0 (λ x14 : ι → ι . λ x15 : ((ι → ι) → ι) → ι . λ x16 . λ x17 : ι → ι . λ x18 . Inj0 0) (x11 0)) (λ x13 . 0))) (x3 (λ x8 . x8) (λ x8 . x8))) ⟶ (∀ x4 . ∀ x5 : (((ι → ι) → ι → ι) → (ι → ι) → ι → ι) → ι . ∀ x6 x7 . x0 (λ x8 : ι → ι . λ x9 : ((ι → ι) → ι) → ι . λ x10 . λ x11 : ι → ι . λ x12 . x12) 0 = Inj0 x6) ⟶ (∀ x4 : (((ι → ι) → ι) → ι) → ι . ∀ x5 . ∀ x6 : (((ι → ι) → ι) → ι) → ι . ∀ x7 . x0 (λ x8 : ι → ι . λ x9 : ((ι → ι) → ι) → ι . λ x10 . λ x11 : ι → ι . λ x12 . 0) (setsum (setsum x5 (x4 (λ x8 : (ι → ι) → ι . x6 (λ x9 : (ι → ι) → ι . 0)))) (Inj0 0)) = x5) ⟶ False.
Apply notE with
0 = 1 leaving 2 subgoals.
The subproof is completed by applying unknownprop_e1454ae89380849e3cb6b4743b200bdcbe47a12d25b42c4b1d68a2f07dac0ac1.
Apply H2 with
0,
λ x4 . 0,
0,
λ x4 . λ x5 : ι → ι . 0,
λ x4 x5 . x4 = 1.
The subproof is completed by applying H2 with 0, λ x4 . 1, 0, λ x4 . λ x5 : ι → ι . 0.