Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type (ι → ο) → ο be given.
Let x3 of type (ι → ο) → ο be given.
Apply pack_c_0_eq with
pack_c 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_c_0_eq2 with x0, x2.
Apply andI with
x0 = x1,
∀ x4 : ι → ο . (∀ x5 . x4 x5 ⟶ x5 ∈ x0) ⟶ x2 x4 = x3 x4 leaving 2 subgoals.
The subproof is completed by applying L2.
Let x4 of type ι → ο be given.
Assume H3: ∀ x5 . x4 x5 ⟶ x5 ∈ x0.
Claim L4: ∀ x5 . x4 x5 ⟶ x5 ∈ x1
Apply L2 with
λ x5 x6 . ∀ x7 . x4 x7 ⟶ x7 ∈ x5.
The subproof is completed by applying H3.
Apply pack_c_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_c (ap x6 1) x4 = x3 x4.
Let x5 of type ο → ο → ο be given.
Apply pack_c_1_eq2 with
x1,
x3,
x4,
λ x6 x7 : ο . x5 x7 x6.
The subproof is completed by applying L4.