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Let x0 of type ι be given.
Apply H0 with λ x1 . x1 = pack_c_u_r_p (ap x1 0) (decode_c (ap x1 1)) (ap (ap x1 2)) (decode_r (ap x1 3)) (decode_p (ap x1 4)).
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 → ι → ο be given.
Let x5 of type ι → ο be given.
Apply pack_c_u_r_p_0_eq2 with x1, x2, x3, x4, x5, λ x6 x7 . pack_c_u_r_p x1 x2 x3 x4 x5 = pack_c_u_r_p x6 (decode_c (ap (pack_c_u_r_p x1 x2 x3 x4 x5) 1)) (ap (ap (pack_c_u_r_p x1 x2 x3 x4 x5) 2)) (decode_r (ap (pack_c_u_r_p x1 x2 x3 x4 x5) 3)) (decode_p (ap (pack_c_u_r_p x1 x2 x3 x4 x5) 4)).
Apply pack_c_u_r_p_ext with x1, x2, decode_c (ap (pack_c_u_r_p x1 x2 x3 x4 x5) 1), x3, ap (ap (pack_c_u_r_p x1 x2 x3 x4 x5) 2), x4, decode_r (ap (pack_c_u_r_p x1 x2 x3 x4 x5) 3), x5, decode_p (ap (pack_c_u_r_p x1 x2 x3 x4 x5) 4) leaving 4 subgoals.
Let x6 of type ι → ο be given.
Assume H2: ∀ x7 . x6 x7 ⟶ x7 ∈ x1.
Apply pack_c_u_r_p_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 H2.
The subproof is completed by applying iff_refl with x2 x6.
The subproof is completed by applying pack_c_u_r_p_2_eq2 with x1, x2, x3, x4, x5.
Let x6 of type ι be given.
Assume H2: x6 ∈ x1.
Let x7 of type ι be given.
Assume H3: x7 ∈ x1.
Apply pack_c_u_r_p_3_eq2 with x1, x2, x3, x4, x5, x6, x7, λ x8 x9 : ο . iff (x4 x6 x7) x8 leaving 3 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
The subproof is completed by applying iff_refl with x4 x6 x7.
Let x6 of type ι be given.
Assume H2: x6 ∈ x1.
Apply pack_c_u_r_p_4_eq2 with x1, x2, x3, x4, x5, x6, λ x7 x8 : ο . iff (x5 x6) x7 leaving 2 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying iff_refl with x5 x6.
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