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.
Let x3 of type ι → ι → ο be given.
Assume H0:
∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . tuple_p (x1 x4) x5 ⟶ ∀ x6 : ι → ι . (∀ x7 . x7 ∈ x1 x4 ⟶ c40a3.. x0 x1 x2 (ap x5 x7) (x6 x7)) ⟶ (∀ x7 . x7 ∈ x1 x4 ⟶ x3 (ap x5 x7) (x6 x7)) ⟶ x3 (lam 2 (λ x7 . If_i (x7 = 0) x4 x5)) (x2 x4 x5 (lam (x1 x4) (λ x7 . x6 x7))).
Let x4 of type ι be given.
Let x5 of type ι be given.
Apply H1 with
λ x6 x7 . and (c40a3.. x0 x1 x2 x6 x7) (x3 x6 x7).
Let x6 of type ι be given.
Assume H2: x6 ∈ x0.
Let x7 of type ι be given.
Let x8 of type ι → ι be given.
Assume H4:
∀ x9 . x9 ∈ x1 x6 ⟶ and (c40a3.. x0 x1 x2 (ap x7 x9) (x8 x9)) (x3 (ap x7 x9) (x8 x9)).
Apply andI with
c40a3.. x0 x1 x2 (lam 2 (λ x9 . If_i (x9 = 0) x6 x7)) (x2 x6 x7 (lam (x1 x6) x8)),
x3 (lam 2 (λ x9 . If_i (x9 = 0) x6 x7)) (x2 x6 x7 (lam (x1 x6) x8)) leaving 2 subgoals.
Apply unknownprop_412f171b3ca442889854b1b6db1f4b4c229650705be86611715b49239feb0217 with
x0,
x1,
x2,
x6,
x7,
x8 leaving 3 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
Let x9 of type ι be given.
Assume H5: x9 ∈ x1 x6.
Apply H4 with
x9,
c40a3.. x0 x1 x2 (ap x7 x9) (x8 x9) leaving 2 subgoals.
The subproof is completed by applying H5.
Assume H6:
c40a3.. x0 x1 x2 (ap x7 x9) (x8 x9).
Assume H7:
x3 (ap x7 x9) (x8 x9).
The subproof is completed by applying H6.
Apply H0 with
x6,
x7,
x8 leaving 4 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
Let x9 of type ι be given.
Assume H5: x9 ∈ x1 x6.
Apply H4 with
x9,
c40a3.. x0 x1 x2 (ap x7 x9) (x8 x9) leaving 2 subgoals.
The subproof is completed by applying H5.
Assume H6:
c40a3.. x0 x1 x2 (ap x7 x9) (x8 x9).
Assume H7:
x3 (ap x7 x9) ....
Let x4 of type ι be given.
Let x5 of type ι be given.
Apply L1 with
x4,
x5,
x3 x4 x5 leaving 2 subgoals.
The subproof is completed by applying H2.
Assume H4: x3 x4 x5.
The subproof is completed by applying H4.