Let x0 of type ι → (ι → ι → ι → ι) → ι → ι → ι be given.
Assume H0: ∀ x1 . ∀ x2 x3 : ι → ι → ι → ι . (∀ x4 . x4 ∈ x1 ⟶ x2 x4 = x3 x4) ⟶ x0 x1 x2 = x0 x1 x3.
Apply In_ind with
λ x1 . In_rec_G_iii x0 x1 (In_rec_iii x0 x1).
Let x1 of type ι be given.
Apply Descr_iii_prop with
In_rec_G_iii x0 x1 leaving 2 subgoals.
Let x2 of type ο be given.
Apply H2 with
x0 x1 (In_rec_iii x0).
Apply In_rec_G_iii_c with
x0,
x1,
In_rec_iii x0.
The subproof is completed by applying H1.
Apply In_rec_G_iii_f with
x0,
x1.
The subproof is completed by applying H0.