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 → ο be given.
Let x4 of type ι → ο be given.
Assume H1: ∀ x5 . x4 x5 ⟶ x5 ∈ x1.
Apply H0 with
λ x5 x6 . x2 x4 = decode_c (ap x6 1) x4.
Apply tuple_3_1_eq with
x1,
encode_c x1 x2,
encode_r x1 x3,
λ x5 x6 . x2 x4 = decode_c x6 x4.
Let x5 of type ο → ο → ο be given.
Apply decode_encode_c with
x1,
x2,
x4,
λ x6 x7 : ο . x5 x7 x6.
The subproof is completed by applying H1.