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 be given.
Assume H0: x3 ∈ x0.
Let x4 of type ι be given.
Let x5 of type ι → ι be given.
Assume H2:
∀ x6 . x6 ∈ x1 x3 ⟶ c40a3.. x0 x1 x2 (ap x4 x6) (x5 x6).
Let x6 of type ι → ι → ο be given.
Assume H3:
∀ 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 H3 with
x3,
x4,
x5 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 H4: x7 ∈ x1 x3.
Apply H2 with
x7,
x6 leaving 2 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H3.