Let x0 of type ι → (ι → ο) → (ι → ο) → ο be given.
Let x1 of type ι be given.
Let x2 of type ι → ο be given.
Let x3 of type ι → ο be given.
Assume H0:
∀ x4 : ι → ο . (∀ x5 . x5 ∈ x1 ⟶ iff (x2 x5) (x4 x5)) ⟶ ∀ x5 : ι → ο . (∀ x6 . x6 ∈ x1 ⟶ iff (x3 x6) (x5 x6)) ⟶ x0 x1 x4 x5 = x0 x1 x2 x3.
Apply pack_p_p_0_eq2 with
x1,
x2,
x3,
λ x4 x5 . x0 x4 (decode_p (ap (pack_p_p x1 x2 x3) 1)) (decode_p (ap (pack_p_p x1 x2 x3) 2)) = x0 x1 x2 x3.
Apply H0 with
decode_p (ap (pack_p_p x1 x2 x3) 1),
decode_p (ap (pack_p_p x1 x2 x3) 2) leaving 2 subgoals.
Let x4 of type ι be given.
Assume H1: x4 ∈ x1.
Apply pack_p_p_1_eq2 with
x1,
x2,
x3,
x4,
λ x5 x6 : ο . iff (x2 x4) x5 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying iff_refl with x2 x4.
Let x4 of type ι be given.
Assume H1: x4 ∈ x1.
Apply pack_p_p_2_eq2 with
x1,
x2,
x3,
x4,
λ x5 x6 : ο . iff (x3 x4) x5 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying iff_refl with x3 x4.