Let x0 of type ι → (ι → ι) → ι be given.
Assume H0:
∀ x1 . ∀ x2 x3 : ι → ι . (∀ x4 . In x4 x1 ⟶ x2 x4 = x3 x4) ⟶ x0 x1 x2 = x0 x1 x3.
Apply unknownprop_acac0f89c78f08b97a9fe27ba4af5f929f74e43a9a77a0beb38d70975279c8b8 with
λ x1 . In_rec_poly_G_i x0 x1 (In_rec_poly_i x0 x1).
Let x1 of type ι be given.
Apply unknownprop_425c67ed3bf75f89891890c077b643570c6e9b6cddd8ad00fbb79dc781422b21 with
λ x2 x3 : (ι → (ι → ι) → ι) → ι → ι . In_rec_poly_G_i x0 x1 (x3 x0 x1).
Apply unknownprop_c3f0de4cb966012957ca752938aa96a32c594389e7aea45227d571c0506618ba with
In_rec_poly_G_i x0 x1,
x0 x1 (In_rec_poly_i x0).
Apply unknownprop_e7fe819b1a8c487858563d302a60bb5640d0c158ddd118b963d5fab9269d2fbb with
x0,
x1,
In_rec_poly_i x0.
The subproof is completed by applying H1.