Let x0 of type ι be given.
Let x1 of type ι → ο be given.
Assume H0:
∀ x2 . x1 x2 ⟶ ∀ x3 . x3 ∈ x2 ⟶ nIn x0 x3.
Let x2 of type ι → ι be given.
Let x3 of type ι → ι be given.
Let x4 of type ι → ι → ι be given.
Let x5 of type ι → ι → ι be given.
Let x6 of type ι be given.
Let x7 of type ι be given.
Apply nat_primrec_S with
1,
λ x8 x9 . CD_mul x0 x1 x2 x3 x4 x5 x6 x9,
x7.
The subproof is completed by applying H1.