Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι → ο be given.
Let x3 of type ι → ο be given.
Apply pack_p_0_eq with
pack_p x0 x2,
x1,
x3.
The subproof is completed by applying H0.
Claim L2: x0 = x1
Apply L1 with
λ x4 x5 . x0 = x5.
The subproof is completed by applying pack_p_0_eq2 with x0, x2.
Apply andI with
x0 = x1,
∀ x4 . x4 ∈ x0 ⟶ x2 x4 = x3 x4 leaving 2 subgoals.
The subproof is completed by applying L2.
Let x4 of type ι be given.
Assume H3: x4 ∈ x0.
Claim L4: x4 ∈ x1
Apply L2 with
λ x5 x6 . x4 ∈ x5.
The subproof is completed by applying H3.
Apply pack_p_1_eq2 with
x0,
x2,
x4,
λ x5 x6 : ο . x6 = x3 x4 leaving 2 subgoals.
The subproof is completed by applying H3.
Apply H0 with
λ x5 x6 . decode_p (ap x6 1) x4 = x3 x4.
Let x5 of type ο → ο → ο be given.
Apply pack_p_1_eq2 with
x1,
x3,
x4,
λ x6 x7 : ο . x5 x7 x6.
The subproof is completed by applying L4.