Let x0 of type ι be given.
Apply H0 with
λ x1 . x1 = pack_b_b_p (ap x1 0) (decode_b (ap x1 1)) (decode_b (ap x1 2)) (decode_p (ap x1 3)).
Let x1 of type ι be given.
Let x2 of type ι → ι → ι be given.
Assume H1: ∀ x3 . x3 ∈ x1 ⟶ ∀ x4 . x4 ∈ x1 ⟶ x2 x3 x4 ∈ x1.
Let x3 of type ι → ι → ι be given.
Assume H2: ∀ x4 . x4 ∈ x1 ⟶ ∀ x5 . x5 ∈ x1 ⟶ x3 x4 x5 ∈ x1.
Let x4 of type ι → ο be given.
Apply pack_b_b_p_0_eq2 with
x1,
x2,
x3,
x4,
λ x5 x6 . pack_b_b_p x1 x2 x3 x4 = pack_b_b_p x5 (decode_b (ap (pack_b_b_p x1 x2 x3 x4) 1)) (decode_b (ap (pack_b_b_p x1 x2 x3 x4) 2)) (decode_p (ap (pack_b_b_p x1 x2 x3 x4) 3)).
Apply pack_b_b_p_ext with
x1,
x2,
decode_b (ap (pack_b_b_p x1 x2 x3 x4) 1),
x3,
decode_b (ap (pack_b_b_p x1 x2 x3 x4) 2),
x4,
decode_p (ap (pack_b_b_p x1 x2 x3 x4) 3) leaving 3 subgoals.
The subproof is completed by applying pack_b_b_p_1_eq2 with x1, x2, x3, x4.
The subproof is completed by applying pack_b_b_p_2_eq2 with x1, x2, x3, x4.
Let x5 of type ι be given.
Assume H3: x5 ∈ x1.
Apply pack_b_b_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 H3.
The subproof is completed by applying iff_refl with x4 x5.