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 ⟶ ∀ x5 x6 . c40a3.. x0 x1 x2 x5 x6 ⟶ x3 = x5 ⟶ x4 = x6.
Let x3 of type ι be given.
Assume H0: x3 ∈ x0.
Let x4 of type ι be given.
Assume H1:
tuple_p ((λ x5 . x1 x5) x3) x4.
Assume H2:
∀ x5 . x5 ∈ (λ x6 . x1 x6) 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 ⟶ ∀ x7 x8 . c40a3.. x0 x1 x2 x7 x8 ⟶ ap x4 x5 = x7 ⟶ x6 = x8.
Claim L4:
∀ x5 x6 . c40a3.. x0 x1 x2 x5 x6 ⟶ x5 = lam 2 (λ x7 . If_i (x7 = 0) x3 x4) ⟶ ∀ x7 x8 . c40a3.. x0 x1 x2 x7 x8 ⟶ x5 = x7 ⟶ x6 = x8Apply unknownprop_37fd53e9c4057a35818d15278ac7ecde5930df6a5519cd34c128a3ae528343a4 with
x0,
x1,
x2,
λ x5 x6 . x5 = lam 2 (λ x7 . If_i (x7 = 0) x3 x4) ⟶ ∀ x7 x8 . c40a3.. x0 x1 x2 x7 x8 ⟶ x5 = x7 ⟶ x6 = x8.
Let x5 of type ι be given.
Assume H4: x5 ∈ x0.
Let x6 of type ι be given.
Let x7 of type ι → ι be given.
Assume H6:
∀ x8 . x8 ∈ x1 x5 ⟶ c40a3.. x0 x1 x2 (ap x6 x8) (x7 x8).
Assume H7:
∀ x8 . x8 ∈ x1 x5 ⟶ ap x6 x8 = lam 2 (λ x9 . If_i (x9 = 0) x3 x4) ⟶ ∀ x9 x10 . c40a3.. x0 x1 x2 x9 x10 ⟶ ap x6 x8 = x9 ⟶ x7 x8 = x10.
Assume H8:
lam 2 (λ x8 . If_i (x8 = 0) x5 x6) = lam 2 (λ x8 . If_i (x8 = 0) x3 x4).
Let x5 of type ι be given.
Apply L4 with
lam 2 (λ x6 . If_i (x6 = 0) x3 x4),
x5 leaving 2 subgoals.
The subproof is completed by applying H5.
Let x6 of type ι → ι → ο be given.
Assume H6:
x6 (lam 2 (λ x7 . If_i (x7 = 0) x3 x4)) (lam 2 (λ x7 . If_i (x7 = 0) x3 x4)).
The subproof is completed by applying H6.