Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι → ο be given.
Assume H0:
x0 = lam 2 (λ x3 . If_i (x3 = 0) x1 (Sep x1 x2)).
Let x3 of type ι be given.
Assume H1: x3 ∈ x1.
Apply H0 with
λ x4 x5 . x2 x3 = decode_p (ap x5 1) x3.
Apply tuple_2_1_eq with
x1,
Sep x1 x2,
λ x4 x5 . x2 x3 = decode_p x5 x3.
Let x4 of type ο → ο → ο be given.
Apply decode_encode_p with
x1,
x2,
x3,
λ x5 x6 : ο . x4 x6 x5.
The subproof is completed by applying H1.