Let x0 of type ι → (ι → ι → ο) → ι → ο be given.
Let x1 of type ι be given.
Let x2 of type ι → ι → ο be given.
Let x3 of type ι be given.
Assume H0:
∀ x4 : ι → ι → ο . (∀ x5 . x5 ∈ x1 ⟶ ∀ x6 . x6 ∈ x1 ⟶ iff (x2 x5 x6) (x4 x5 x6)) ⟶ x0 x1 x4 x3 = x0 x1 x2 x3.
Apply pack_r_e_0_eq2 with
x1,
x2,
x3,
λ x4 x5 . x0 x4 (decode_r (ap (pack_r_e x1 x2 x3) 1)) (ap (pack_r_e x1 x2 x3) 2) = x0 x1 x2 x3.
Apply pack_r_e_2_eq2 with
x1,
x2,
x3,
λ x4 x5 . x0 x1 (decode_r (ap (pack_r_e x1 x2 x3) 1)) x4 = x0 x1 x2 x3.
Apply H0 with
decode_r (ap (pack_r_e x1 x2 x3) 1).
Let x4 of type ι be given.
Assume H1: x4 ∈ x1.
Let x5 of type ι be given.
Assume H2: x5 ∈ x1.
Apply pack_r_e_1_eq2 with
x1,
x2,
x3,
x4,
x5,
λ x6 x7 : ο . iff (x2 x4 x5) x6 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 x2 x4 x5.