Let x0 of type ι be given.
Apply H0 with
λ x1 . x1 = pack_u_r (ap x1 0) (ap (ap x1 1)) (decode_r (ap x1 2)).
Let x1 of type ι be given.
Let x2 of type ι → ι be given.
Assume H1: ∀ x3 . x3 ∈ x1 ⟶ x2 x3 ∈ x1.
Let x3 of type ι → ι → ο be given.
Apply pack_u_r_0_eq2 with
x1,
x2,
x3,
λ x4 x5 . pack_u_r x1 x2 x3 = pack_u_r x4 (ap (ap (pack_u_r x1 x2 x3) 1)) (decode_r (ap (pack_u_r x1 x2 x3) 2)).
Apply pack_u_r_ext with
x1,
x2,
ap (ap (pack_u_r x1 x2 x3) 1),
x3,
decode_r (ap (pack_u_r x1 x2 x3) 2) leaving 2 subgoals.
The subproof is completed by applying pack_u_r_1_eq2 with x1, x2, x3.
Let x4 of type ι be given.
Assume H2: x4 ∈ x1.
Let x5 of type ι be given.
Assume H3: x5 ∈ x1.
Apply pack_u_r_2_eq2 with
x1,
x2,
x3,
x4,
x5,
λ x6 x7 : ο . iff (x3 x4 x5) x6 leaving 3 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
The subproof is completed by applying iff_refl with x3 x4 x5.