Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type (ι → ο) → ο be given.
Let x3 of type ι → ι be given.
Let x4 of type ι → ο be given.
Let x5 of type ι be given.
Let x6 of type ι be given.
Assume H1: x6 ∈ x1.
Apply H0 with
λ x7 x8 . x4 x6 = decode_p (ap x8 3) x6.
Apply tuple_5_3_eq with
x1,
encode_c x1 x2,
lam x1 x3,
Sep x1 x4,
x5,
λ x7 x8 . x4 x6 = decode_p x8 x6.
Let x7 of type ο → ο → ο be given.
Apply decode_encode_p with
x1,
x4,
x6,
λ x8 x9 : ο . x7 x9 x8.
The subproof is completed by applying H1.