Let x0 of type ι be given.
Apply H0 with
λ x1 . x1 = pack_c_u_p (ap x1 0) (decode_c (ap x1 1)) (ap (ap x1 2)) (decode_p (ap x1 3)).
Let x1 of type ι be given.
Let x2 of type (ι → ο) → ο be given.
Let x3 of type ι → ι be given.
Assume H1: ∀ x4 . x4 ∈ x1 ⟶ x3 x4 ∈ x1.
Let x4 of type ι → ο be given.
Apply pack_c_u_p_0_eq2 with
x1,
x2,
x3,
x4,
λ x5 x6 . pack_c_u_p x1 x2 x3 x4 = pack_c_u_p x5 (decode_c (ap (pack_c_u_p x1 x2 x3 x4) 1)) (ap (ap (pack_c_u_p x1 x2 x3 x4) 2)) (decode_p (ap (pack_c_u_p x1 x2 x3 x4) 3)).
Apply pack_c_u_p_ext with
x1,
x2,
decode_c (ap (pack_c_u_p x1 x2 x3 x4) 1),
x3,
ap (ap (pack_c_u_p x1 x2 x3 x4) 2),
x4,
decode_p (ap (pack_c_u_p x1 x2 x3 x4) 3) leaving 3 subgoals.
Let x5 of type ι → ο be given.
Assume H2: ∀ x6 . x5 x6 ⟶ x6 ∈ x1.
Apply pack_c_u_p_1_eq2 with
x1,
x2,
x3,
x4,
x5,
λ x6 x7 : ο . iff (x2 x5) x6 leaving 2 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying iff_refl with x2 x5.
The subproof is completed by applying pack_c_u_p_2_eq2 with x1, x2, x3, x4.
Let x5 of type ι be given.
Assume H2: x5 ∈ x1.
Apply pack_c_u_p_3_eq2 with
x1,
x2,
x3,
x4,
x5,
λ x6 x7 : ο . iff (x4 x5) x6 leaving 2 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying iff_refl with x4 x5.