Let x0 of type ι → (ι → ι → ι) → ι → ι be given.
Assume H0:
∀ x1 . ∀ x2 x3 : ι → ι → ι . (∀ x4 . prim1 x4 x1 ⟶ x2 x4 = x3 x4) ⟶ x0 x1 x2 = x0 x1 x3.
Let x1 of type ι be given.
Let x2 of type ι → (ι → ι) → ο be given.
Assume H1:
∀ x3 . ∀ x4 : ι → ι → ι . (∀ x5 . prim1 x5 x3 ⟶ x2 x5 (x4 x5)) ⟶ x2 x3 (x0 x3 x4).
Apply H1 with
x1,
In_rec_ii x0.
Let x3 of type ι be given.
Apply In_rec_G_ii_In_rec_ii with
x0,
x3,
x2 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.