Let x0 of type ι be given.
Let x1 of type ι → ι be given.
Let x2 of type ι be given.
Assume H0:
x2 ∈ Pi x0 (λ x3 . x1 x3).
Let x3 of type ι be given.
Assume H1:
x3 ∈ Pi x0 (λ x4 . x1 x4).
Assume H2:
∀ x4 . x4 ∈ x0 ⟶ ap x2 x4 ⊆ ap x3 x4.
Apply PiE with
x0,
x1,
x2,
x2 ⊆ x3 leaving 2 subgoals.
The subproof is completed by applying H0.
Assume H3:
∀ x4 . x4 ∈ x2 ⟶ and (pair_p x4) (ap x4 0 ∈ x0).
Assume H4:
∀ x4 . x4 ∈ x0 ⟶ ap x2 x4 ∈ x1 x4.
Apply PiE with
x0,
x1,
x3,
x2 ⊆ x3 leaving 2 subgoals.
The subproof is completed by applying H1.
Assume H5:
∀ x4 . x4 ∈ x3 ⟶ and (pair_p x4) (ap x4 0 ∈ x0).
Assume H6:
∀ x4 . x4 ∈ x0 ⟶ ap x3 x4 ∈ x1 x4.
Let x4 of type ι be given.
Assume H7: x4 ∈ x2.
Apply H3 with
x4,
x4 ∈ x3 leaving 2 subgoals.
The subproof is completed by applying H7.
Apply H8 with
λ x5 x6 . x5 ∈ x3.
Apply apE with
x3,
ap x4 0,
ap x4 1.
Apply H2 with
ap x4 0,
ap x4 1 leaving 2 subgoals.
The subproof is completed by applying H9.
Apply apI with
x2,
ap x4 0,
ap x4 1.
Apply H8 with
λ x5 x6 . x6 ∈ x2.
The subproof is completed by applying H7.