Let x0 of type ι → (ι → ι → ο) → (ι → ι → ο) → (ι → ο) → ο be given.
Let x1 of type ι be given.
Let x2 of type ι → ι → ο be given.
Let x3 of type ι → ι → ο be given.
Let x4 of type ι → ο be given.
Assume H0:
∀ x5 : ι → ι → ο . (∀ x6 . x6 ∈ x1 ⟶ ∀ x7 . x7 ∈ x1 ⟶ iff (x2 x6 x7) (x5 x6 x7)) ⟶ ∀ x6 : ι → ι → ο . (∀ x7 . x7 ∈ x1 ⟶ ∀ x8 . x8 ∈ x1 ⟶ iff (x3 x7 x8) (x6 x7 x8)) ⟶ ∀ x7 : ι → ο . (∀ x8 . x8 ∈ x1 ⟶ iff (x4 x8) (x7 x8)) ⟶ x0 x1 x5 x6 x7 = x0 x1 x2 x3 x4.
Apply pack_r_r_p_0_eq2 with
x1,
x2,
x3,
x4,
λ x5 x6 . x0 x5 (decode_r (ap (pack_r_r_p x1 x2 x3 x4) 1)) (decode_r (ap (pack_r_r_p x1 x2 x3 x4) 2)) (decode_p (ap (pack_r_r_p x1 x2 x3 x4) 3)) = x0 x1 x2 x3 x4.
Apply H0 with
decode_r (ap (pack_r_r_p x1 x2 x3 x4) 1),
decode_r (ap (pack_r_r_p x1 x2 x3 x4) 2),
decode_p (ap (pack_r_r_p x1 x2 x3 x4) 3) leaving 3 subgoals.
Let x5 of type ι be given.
Assume H1: x5 ∈ x1.
Let x6 of type ι be given.
Assume H2: x6 ∈ x1.
Apply pack_r_r_p_1_eq2 with
x1,
x2,
x3,
x4,
x5,
x6,
λ x7 x8 : ο . iff (x2 x5 x6) x7 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 x5 x6.
Let x5 of type ι be given.
Assume H1: x5 ∈ x1.
Let x6 of type ι be given.
Assume H2: x6 ∈ x1.
Apply pack_r_r_p_2_eq2 with
x1,
x2,
x3,
x4,
x5,
x6,
λ x7 x8 : ο . iff (x3 x5 x6) x7 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 x5 x6.
Let x5 of type ι be given.
Assume H1: x5 ∈ x1.
Apply pack_r_r_p_3_eq2 with
x1,
x2,
x3,
x4,
x5,
λ x6 x7 : ο . iff (x4 x5) x6 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying iff_refl with x4 x5.