Let x0 of type ι be given.
Apply H0 with
λ x1 . x1 = pack_b_r_e (ap x1 0) (decode_b (ap x1 1)) (decode_r (ap x1 2)) (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 → ο be given.
Let x4 of type ι be given.
Assume H2: x4 ∈ x1.
Apply pack_b_r_e_0_eq2 with
x1,
x2,
x3,
x4,
λ x5 x6 . pack_b_r_e x1 x2 x3 x4 = pack_b_r_e x5 (decode_b (ap (pack_b_r_e x1 x2 x3 x4) 1)) (decode_r (ap (pack_b_r_e x1 x2 x3 x4) 2)) (ap (pack_b_r_e x1 x2 x3 x4) 3).
Apply pack_b_r_e_3_eq2 with
x1,
x2,
x3,
x4,
λ x5 x6 . pack_b_r_e x1 x2 x3 x4 = pack_b_r_e x1 (decode_b (ap (pack_b_r_e x1 x2 x3 x4) 1)) (decode_r (ap (pack_b_r_e x1 x2 x3 x4) 2)) x5.
Apply pack_b_r_e_ext with
x1,
x2,
decode_b (ap (pack_b_r_e x1 x2 x3 x4) 1),
x3,
decode_r (ap (pack_b_r_e x1 x2 x3 x4) 2),
x4 leaving 2 subgoals.
The subproof is completed by applying pack_b_r_e_1_eq2 with x1, x2, x3, x4.
Let x5 of type ι be given.
Assume H3: x5 ∈ x1.
Let x6 of type ι be given.
Assume H4: x6 ∈ x1.
Apply pack_b_r_e_2_eq2 with
x1,
x2,
x3,
x4,
x5,
x6,
λ x7 x8 : ο . iff (x3 x5 x6) x7 leaving 3 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
The subproof is completed by applying iff_refl with x3 x5 x6.