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.
Let x4 of type ι → ι be given.
Let x5 of type ι → ι be given.
Apply pack_c_u_0_eq with
pack_c_u x0 x2 x4,
x1,
x3,
x5.
The subproof is completed by applying H0.
Claim L2: x0 = x1
Apply L1 with
λ x6 x7 . x0 = x7.
The subproof is completed by applying pack_c_u_0_eq2 with x0, x2, x4.
Apply and3I with
x0 = x1,
∀ x6 : ι → ο . (∀ x7 . x6 x7 ⟶ x7 ∈ x0) ⟶ x2 x6 = x3 x6,
∀ x6 . x6 ∈ x0 ⟶ x4 x6 = x5 x6 leaving 3 subgoals.
The subproof is completed by applying L2.
Let x6 of type ι → ο be given.
Assume H3: ∀ x7 . x6 x7 ⟶ x7 ∈ x0.
Apply pack_c_u_1_eq2 with
x0,
x2,
x4,
x6,
λ x7 x8 : ο . x8 = x3 x6 leaving 2 subgoals.
The subproof is completed by applying H3.
Claim L4: ∀ x7 . x6 x7 ⟶ x7 ∈ x1
Apply L2 with
λ x7 x8 . ∀ x9 . x6 x9 ⟶ x9 ∈ x7.
The subproof is completed by applying H3.
Apply H0 with
λ x7 x8 . decode_c (ap x8 1) x6 = x3 x6.
Let x7 of type ο → ο → ο be given.
Apply pack_c_u_1_eq2 with
x1,
x3,
x5,
x6,
λ x8 x9 : ο . x7 x9 x8.
The subproof is completed by applying L4.
Let x6 of type ι be given.
Assume H3: x6 ∈ x0.
Apply pack_c_u_2_eq2 with
x0,
x2,
x4,
x6,
λ x7 x8 . x8 = x5 x6 leaving 2 subgoals.
The subproof is completed by applying H3.
Claim L4: x6 ∈ x1
Apply L2 with
λ x7 x8 . x6 ∈ x7.
The subproof is completed by applying H3.
Apply H0 with
λ x7 x8 . ap (ap x8 2) x6 = x5 x6.
Let x7 of type ι → ι → ο be given.
Apply pack_c_u_2_eq2 with
x1,
x3,
x5,
x6,
λ x8 x9 . x7 x9 x8.
The subproof is completed by applying L4.