Let x0 of type ι be given.
Let x1 of type ι be given.
Apply encode_u_ext with
omega,
λ x2 . nat_primrec x0 (λ x3 x4 . ap x1 x3) x2,
λ x2 . ap (4ec03.. x0 x1) x2.
Let x2 of type ι be given.
Let x3 of type ι → ι → ο be given.
Apply beta with
omega,
λ x4 . nat_primrec x0 (λ x5 x6 . ap x1 x5) x4,
x2,
λ x4 x5 . x3 x5 x4.
The subproof is completed by applying H0.