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.

Let x5 of type ι → ι be given.

Assume H0: x0 = pack_b_b_u_u x1 x2 x3 x4 x5.

Let x6 of type ι be given.

Assume H1: x6 ∈ x1.

Apply H0 with λ x7 x8 . x4 x6 = ap (ap x8 3) x6.

Apply tuple_5_3_eq with x1, encode_b x1 x2, encode_b x1 x3, lam x1 x4, lam x1 x5, λ x7 x8 . x4 x6 = ap x8 x6.

Let x7 of type ι → ι → ο be given.

Apply beta with x1, x4, x6, λ x8 x9 . x7 x9 x8.

The subproof is completed by applying H1.

■

Proofgold Proof