Let x0 of type ι → (ι → ι) → ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Apply unknownprop_7461507986cbf49e907e297fd5f04e13d3ceb8c3463464280e794a94eb50b6a5 with
λ x3 x4 : (ι → (ι → ι) → ι) → ι → ι → ο . x3 x0 x1 x2,
λ x3 x4 . ∃ x5 : ι → ι . and (∀ x6 . In x6 x3 ⟶ In_rec_poly_G_i x0 x6 (x5 x6)) (x4 = x0 x3 x5) leaving 2 subgoals.
The subproof is completed by applying H0.
Let x3 of type ι be given.
Let x4 of type ι → ι be given.
Assume H1:
∀ x5 . In x5 x3 ⟶ (λ x6 x7 . ∃ x8 : ι → ι . and (∀ x9 . In x9 x6 ⟶ In_rec_poly_G_i x0 x9 (x8 x9)) (x7 = x0 x6 x8)) x5 (x4 x5).
Let x5 of type ο be given.
Assume H2:
∀ x6 : ι → ι . and (∀ x7 . In x7 x3 ⟶ In_rec_poly_G_i x0 x7 (x6 x7)) (x0 x3 x4 = x0 x3 x6) ⟶ x5.
Apply H2 with
x4.
Apply unknownprop_389e2fb1855352fcc964ea44fe6723d7a1c2d512f04685300e3e97621725b977 with
∀ x6 . In x6 x3 ⟶ In_rec_poly_G_i x0 x6 (x4 x6),
x0 x3 x4 = x0 x3 x4 leaving 2 subgoals.
Let x6 of type ι be given.
Apply unknownprop_1282a4db08642dec5b9520afa7e7bcc65abab8c280c90320dfb50ea6b896a61f with
λ x7 : ι → ι . ∀ x8 . In x8 x6 ⟶ In_rec_poly_G_i x0 x8 (x7 x8),
λ x7 : ι → ι . x4 x6 = x0 x6 x7,
In_rec_poly_G_i x0 x6 (x4 x6) leaving 2 subgoals.
Apply H1 with
x6.
The subproof is completed by applying H3.
Let x7 of type ι → ι be given.
Assume H5: x4 x6 = x0 x6 x7.
Apply H5 with
λ x8 x9 . In_rec_poly_G_i x0 x6 x9.
Apply unknownprop_e7fe819b1a8c487858563d302a60bb5640d0c158ddd118b963d5fab9269d2fbb with
x0,
x6,
x7.
The subproof is completed by applying H4.
Let x6 of type ι → ι → ο be given.
Assume H3: x6 (x0 x3 x4) (x0 x3 x4).
The subproof is completed by applying H3.