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Let x0 of type ι be given.
Apply H0 with λ x1 . x1 = pack_p_p_e_e (ap x1 0) (decode_p (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 → ο be given.
Let x3 of type ι → ο be given.
Let x4 of type ι be given.
Assume H1: x4 ∈ x1.
Let x5 of type ι be given.
Assume H2: x5 ∈ x1.
Apply pack_p_p_e_e_0_eq2 with x1, x2, x3, x4, x5, λ x6 x7 . pack_p_p_e_e x1 x2 x3 x4 x5 = pack_p_p_e_e x6 (decode_p (ap (pack_p_p_e_e x1 x2 x3 x4 x5) 1)) (decode_p (ap (pack_p_p_e_e x1 x2 x3 x4 x5) 2)) (ap (pack_p_p_e_e x1 x2 x3 x4 x5) 3) (ap (pack_p_p_e_e x1 x2 x3 x4 x5) 4).
Apply pack_p_p_e_e_3_eq2 with x1, x2, x3, x4, x5, λ x6 x7 . pack_p_p_e_e x1 x2 x3 x4 x5 = pack_p_p_e_e x1 (decode_p (ap (pack_p_p_e_e x1 x2 x3 x4 x5) 1)) (decode_p (ap (pack_p_p_e_e x1 x2 x3 x4 x5) 2)) x6 (ap (pack_p_p_e_e x1 x2 x3 x4 x5) 4).
Apply pack_p_p_e_e_4_eq2 with x1, x2, x3, x4, x5, λ x6 x7 . pack_p_p_e_e x1 x2 x3 x4 x5 = pack_p_p_e_e x1 (decode_p (ap (pack_p_p_e_e x1 x2 x3 x4 x5) 1)) (decode_p (ap (pack_p_p_e_e x1 x2 x3 x4 x5) 2)) x4 x6.
Apply pack_p_p_e_e_ext with x1, x2, decode_p (ap (pack_p_p_e_e x1 x2 x3 x4 x5) 1), x3, decode_p (ap (pack_p_p_e_e x1 x2 x3 x4 x5) 2), x4, x5 leaving 2 subgoals.
Let x6 of type ι be given.
Assume H3: x6 ∈ x1.
Apply pack_p_p_e_e_1_eq2 with x1, x2, x3, x4, x5, x6, λ x7 x8 : ο . iff (x2 x6) x7 leaving 2 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying iff_refl with x2 x6.
Let x6 of type ι be given.
Assume H3: x6 ∈ x1.
Apply pack_p_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 H3.
The subproof is completed by applying iff_refl with x3 x6.
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