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).
Claim L1:
lam x0 (λ x3 . ap x2 x3) ∈ Pi x0 (λ x3 . x1 x3)Apply lam_Pi with
x0,
λ x3 . x1 x3,
λ x3 . ap x2 x3.
Let x3 of type ι be given.
Assume H1: x3 ∈ x0.
Apply ap_Pi with
x0,
x1,
x2,
x3 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply Pi_ext with
x0,
x1,
lam x0 (λ x3 . ap x2 x3),
x2 leaving 3 subgoals.
The subproof is completed by applying L1.
The subproof is completed by applying H0.
Let x3 of type ι be given.
Assume H2: x3 ∈ x0.
Apply beta with
x0,
ap x2,
x3.
The subproof is completed by applying H2.