Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι → ι → ο be given.
Let x3 of type ι → ι → ο be given.
Apply pack_r_0_eq with
pack_r 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_r_0_eq2 with x0, x2.
Apply andI with
x0 = x1,
∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x2 x4 x5 = x3 x4 x5 leaving 2 subgoals.
The subproof is completed by applying L2.
Let x4 of type ι be given.
Assume H3: x4 ∈ x0.
Let x5 of type ι be given.
Assume H4: x5 ∈ x0.
Claim L5: x4 ∈ x1
Apply L2 with
λ x6 x7 . x4 ∈ x6.
The subproof is completed by applying H3.
Claim L6: x5 ∈ x1
Apply L2 with
λ x6 x7 . x5 ∈ x6.
The subproof is completed by applying H4.
Apply pack_r_1_eq2 with
x0,
x2,
x4,
x5,
λ x6 x7 : ο . x7 = x3 x4 x5 leaving 3 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
Apply H0 with
λ x6 x7 . decode_r (ap x7 1) x4 x5 = x3 x4 x5.
Let x6 of type ο → ο → ο be given.
Apply pack_r_1_eq2 with
x1,
x3,
x4,
x5,
λ x7 x8 : ο . x6 x8 x7 leaving 2 subgoals.
The subproof is completed by applying L5.
The subproof is completed by applying L6.