Let x0 of type ι be given.

Let x1 of type ι be given.

Let x2 of type ι → ι → ι be given.

Let x3 of type ι → ι → ι be given.

Let x4 of type ι → ι be given.

Assume H0: x0 = pack_b_b_u x1 x2 x3 x4.

Let x5 of type ι be given.

Assume H1: x5 ∈ x1.

Apply H0 with λ x6 x7 . x4 x5 = ap (ap x7 3) x5.

Apply tuple_4_3_eq with x1, encode_b x1 x2, encode_b x1 x3, lam x1 x4, λ x6 x7 . x4 x5 = ap x7 x5.

Let x6 of type ι → ι → ο be given.

Apply beta with x1, x4, x5, λ x7 x8 . x6 x8 x7.

The subproof is completed by applying H1.

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Proofgold Proof