Let x0 of type ι be given.
Apply H0 with
λ x1 . x1 = pack_u_p_e_e (ap x1 0) (ap (ap x1 1)) (decode_p (ap x1 2)) (ap x1 3) (ap x1 4).
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 → ο be given.
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_p_e_e_0_eq2 with
x1,
x2,
x3,
x4,
x5,
λ x6 x7 . pack_u_p_e_e x1 x2 x3 x4 x5 = pack_u_p_e_e x6 (ap (ap (pack_u_p_e_e x1 x2 x3 x4 x5) 1)) (decode_p (ap (pack_u_p_e_e x1 x2 x3 x4 x5) 2)) (ap (pack_u_p_e_e x1 x2 x3 x4 x5) 3) (ap (pack_u_p_e_e x1 x2 x3 x4 x5) 4).
Apply pack_u_p_e_e_3_eq2 with
x1,
x2,
x3,
x4,
x5,
λ x6 x7 . pack_u_p_e_e x1 x2 x3 x4 x5 = pack_u_p_e_e x1 (ap (ap (pack_u_p_e_e x1 x2 x3 x4 x5) 1)) (decode_p (ap (pack_u_p_e_e x1 x2 x3 x4 x5) 2)) x6 (ap (pack_u_p_e_e x1 x2 x3 x4 x5) 4).
Apply pack_u_p_e_e_4_eq2 with
x1,
x2,
x3,
x4,
x5,
λ x6 x7 . pack_u_p_e_e x1 x2 x3 x4 x5 = pack_u_p_e_e x1 (ap (ap (pack_u_p_e_e x1 x2 x3 x4 x5) 1)) (decode_p (ap (pack_u_p_e_e x1 x2 x3 x4 x5) 2)) x4 x6.
Apply pack_u_p_e_e_ext with
x1,
x2,
ap (ap (pack_u_p_e_e x1 x2 x3 x4 x5) 1),
x3,
decode_p (ap (pack_u_p_e_e x1 x2 x3 x4 x5) 2),
x4,
x5 leaving 2 subgoals.
The subproof is completed by applying pack_u_p_e_e_1_eq2 with x1, x2, x3, x4, x5.
Let x6 of type ι be given.
Assume H4: x6 ∈ x1.
Apply pack_u_p_e_e_2_eq2 with
x1,
x2,
x3,
x4,
x5,
x6,
λ x7 x8 : ο . iff (x3 x6) x7 leaving 2 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying iff_refl with x3 x6.