Let x0 of type ι be given.
Let x1 of type ι → ι be given.
Let x2 of type ι → ι → ι → ι be given.
Apply unknownprop_65880fc9b48bb00c7fa40a7cbb81970d0089e79d340752d8478eec02a585f2ec with
x0,
λ x3 . x1 x3,
λ x3 . ∃ x4 . c40a3.. x0 x1 x2 x3 x4.
Let x3 of type ι be given.
Assume H0: x3 ∈ x0.
Let x4 of type ι be given.
Assume H2:
∀ x5 . x5 ∈ x1 x3 ⟶ ap x4 x5 ∈ c8f46.. x0 (λ x6 . x1 x6).
Assume H3:
∀ x5 . x5 ∈ x1 x3 ⟶ ∃ x6 . c40a3.. x0 x1 x2 (ap x4 x5) x6.
Claim L4:
∀ x5 . x5 ∈ x1 x3 ⟶ c40a3.. x0 x1 x2 (ap x4 x5) ((λ x6 . prim0 (λ x7 . c40a3.. x0 x1 x2 (ap x4 x6) x7)) x5)Let x5 of type ι be given.
Assume H4: x5 ∈ x1 x3.
Apply Eps_i_ex with
c40a3.. x0 x1 x2 (ap x4 x5).
Apply H3 with
x5.
The subproof is completed by applying H4.
Let x5 of type ο be given.
Assume H5:
∀ x6 . c40a3.. x0 x1 x2 (lam 2 (λ x7 . If_i (x7 = 0) x3 x4)) x6 ⟶ x5.
Apply H5 with
x2 x3 x4 (lam (x1 x3) (λ x6 . (λ x7 . prim0 (λ x8 . c40a3.. x0 x1 x2 (ap x4 x7) x8)) x6)).
Let x6 of type ι → ι → ο be given.
Assume H6:
∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . tuple_p (x1 x7) x8 ⟶ ∀ x9 : ι → ι . (∀ x10 . x10 ∈ x1 x7 ⟶ x6 (ap x8 x10) (x9 x10)) ⟶ x6 (lam 2 (λ x10 . If_i (x10 = 0) x7 x8)) (x2 x7 x8 (lam (x1 x7) x9)).
Apply H6 with
x3,
x4,
λ x7 . prim0 (λ x8 . c40a3.. x0 x1 x2 (ap x4 x7) x8) leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Let x7 of type ι be given.
Assume H7: x7 ∈ x1 x3.
Apply L4 with
x7,
x6 leaving 2 subgoals.
The subproof is completed by applying H7.
The subproof is completed by applying H6.