Let x0 of type ι → (ι → ι) → (ι → ι → ο) → ι → ι → ο be given.
Let x1 of type ι be given.
Let x2 of type ι → ι be given.
Let x3 of type ι → ι → ο be given.
Let x4 of type ι be given.
Let x5 of type ι be given.
Assume H0:
∀ x6 : ι → ι . (∀ x7 . x7 ∈ x1 ⟶ x2 x7 = x6 x7) ⟶ ∀ x7 : ι → ι → ο . (∀ x8 . x8 ∈ x1 ⟶ ∀ x9 . x9 ∈ x1 ⟶ iff (x3 x8 x9) (x7 x8 x9)) ⟶ x0 x1 x6 x7 x4 x5 = x0 x1 x2 x3 x4 x5.
Apply pack_u_r_e_e_0_eq2 with
x1,
x2,
x3,
x4,
x5,
λ x6 x7 . x0 x6 (ap (ap (pack_u_r_e_e x1 x2 x3 x4 x5) 1)) (decode_r (ap (pack_u_r_e_e x1 x2 x3 x4 x5) 2)) (ap (pack_u_r_e_e x1 x2 x3 x4 x5) 3) (ap (pack_u_r_e_e x1 x2 x3 x4 x5) 4) = x0 x1 x2 x3 x4 x5.
Apply pack_u_r_e_e_3_eq2 with
x1,
x2,
x3,
x4,
x5,
λ x6 x7 . x0 x1 (ap (ap (pack_u_r_e_e x1 x2 x3 x4 x5) 1)) (decode_r (ap (pack_u_r_e_e x1 x2 x3 x4 x5) 2)) x6 (ap (pack_u_r_e_e x1 x2 x3 x4 x5) 4) = x0 x1 x2 x3 x4 x5.
Apply pack_u_r_e_e_4_eq2 with
x1,
x2,
x3,
x4,
x5,
λ x6 x7 . x0 x1 (ap (ap (pack_u_r_e_e x1 x2 x3 x4 x5) 1)) (decode_r (ap (pack_u_r_e_e x1 x2 x3 x4 x5) 2)) x4 x6 = x0 x1 x2 x3 x4 x5.
Apply H0 with
ap (ap (pack_u_r_e_e x1 x2 x3 x4 x5) 1),
decode_r (ap (pack_u_r_e_e x1 x2 x3 x4 x5) 2) leaving 2 subgoals.
The subproof is completed by applying pack_u_r_e_e_1_eq2 with x1, x2, x3, x4, x5.
Let x6 of type ι be given.
Assume H1: x6 ∈ x1.
Let x7 of type ι be given.
Assume H2: x7 ∈ x1.
Apply pack_u_r_e_e_2_eq2 with
x1,
x2,
x3,
x4,
x5,
x6,
x7,
λ x8 x9 : ο . iff (x3 x6 x7) x8 leaving 3 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
The subproof is completed by applying iff_refl with x3 x6 x7.