Let x0 of type ι → (ι → ι) → ι be given.
Assume H0:
∀ x1 . ∀ x2 x3 : ι → ι . (∀ x4 . prim1 x4 x1 ⟶ x2 x4 = x3 x4) ⟶ x0 x1 x2 = x0 x1 x3.
Apply In_ind with
λ x1 . In_rec_i_G x0 x1 (In_rec_i x0 x1).
Let x1 of type ι be given.
Apply Eps_i_ax with
In_rec_i_G x0 x1,
x0 x1 (In_rec_i x0).
Apply In_rec_i_G_c with
x0,
x1,
In_rec_i x0.
The subproof is completed by applying H1.