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.
Let x1 of type ι be given.
Apply unknownprop_7461507986cbf49e907e297fd5f04e13d3ceb8c3463464280e794a94eb50b6a5 with
λ x2 x3 : (ι → (ι → ι) → ι) → ι → ι → ο . x3 x0 x1 (x0 x1 (In_rec_poly_i x0)).
Let x2 of type ι → ι → ο be given.
Assume H1:
∀ x3 . ∀ x4 : ι → ι . (∀ x5 . In x5 x3 ⟶ x2 x5 (x4 x5)) ⟶ x2 x3 (x0 x3 x4).
Apply H1 with
x1,
In_rec_poly_i x0.
Let x3 of type ι be given.
Apply unknownprop_7461507986cbf49e907e297fd5f04e13d3ceb8c3463464280e794a94eb50b6a5 with
λ x4 x5 : (ι → (ι → ι) → ι) → ι → ι → ο . x4 x0 x3 (In_rec_poly_i x0 x3),
x2 leaving 2 subgoals.
Apply unknownprop_ea667e8c5984fa10a111dd451427cddccf3b49b0ad2764e849c70aa538387c01 with
x0,
x3.
The subproof is completed by applying H0.
The subproof is completed by applying H1.