Let x0 of type ι be given.

Let x1 of type ι be given.

Let x2 of type (ι → ο) → ο be given.

Let x3 of type (ι → ο) → ο be given.

Let x4 of type ι → ι → ο be given.

Let x5 of type ι → ι → ο be given.

Let x6 of type ι → ο be given.

Let x7 of type ι → ο be given.

Let x8 of type ι → ο be given.

Let x9 of type ι → ο be given.

Assume H0: pack_c_r_p_p x0 x2 x4 x6 x8 = pack_c_r_p_p x1 x3 x5 x7 x9.

Claim L1: x1 = ap (pack_c_r_p_p x0 x2 x4 ... ...) 0

...

Claim L2: x0 = x1

Apply L1 with λ x10 x11 . x0 = x11.

The subproof is completed by applying pack_c_r_p_p_0_eq2 with x0, x2, x4, x6, x8.

Apply and5I with x0 = x1, ∀ x10 : ι → ο . (∀ x11 . x10 x11 ⟶ x11 ∈ x0) ⟶ x2 x10 = x3 x10, ∀ x10 . x10 ∈ x0 ⟶ ∀ x11 . x11 ∈ x0 ⟶ x4 x10 x11 = x5 x10 x11, ∀ x10 . x10 ∈ x0 ⟶ x6 x10 = x7 x10, ∀ x10 . x10 ∈ x0 ⟶ x8 x10 = x9 x10 leaving 5 subgoals.

The subproof is completed by applying L2.

Let x10 of type ι → ο be given.

Assume H3: ∀ x11 . x10 x11 ⟶ x11 ∈ x0.

Apply pack_c_r_p_p_1_eq2 with x0, x2, x4, x6, x8, x10, λ x11 x12 : ο . x12 = x3 x10 leaving 2 subgoals.

The subproof is completed by applying H3.

Claim L4: ∀ x11 . x10 x11 ⟶ x11 ∈ x1

Apply L2 with λ x11 x12 . ∀ x13 . x10 x13 ⟶ x13 ∈ x11.

The subproof is completed by applying H3.

Apply H0 with λ x11 x12 . decode_c (ap x12 1) x10 = x3 x10.

Let x11 of type ο → ο → ο be given.

Apply pack_c_r_p_p_1_eq2 with x1, x3, x5, x7, x9, x10, λ x12 x13 : ο . x11 x13 x12.

The subproof is completed by applying L4.

Let x10 of type ι be given.

Assume H3: x10 ∈ x0.

Let x11 of type ι be given.

Assume H4: x11 ∈ x0.

Apply pack_c_r_p_p_2_eq2 with x0, x2, x4, x6, x8, x10, x11, λ x12 x13 : ο . x13 = x5 x10 x11 leaving 3 subgoals.

The subproof is completed by applying H3.

The subproof is completed by applying H4.

Claim L5: x10 ∈ x1

Apply L2 with λ x12 x13 . x10 ∈ x12.

The subproof is completed by applying H3.

Claim L6: x11 ∈ x1

Apply L2 with λ x12 x13 . x11 ∈ x12.

The subproof is completed by applying H4.

Apply H0 with λ x12 x13 . decode_r (ap x13 2) x10 x11 = x5 x10 x11.

Let x12 of type ο → ο → ο be given.

Apply pack_c_r_p_p_2_eq2 with x1, x3, x5, x7, x9, x10, x11, λ x13 x14 : ο . x12 x14 x13 leaving 2 subgoals.

The subproof is completed by applying L5.

The subproof is completed by applying L6.

Let x10 of type ι be given.

Assume H3: x10 ∈ x0.

Apply pack_c_r_p_p_3_eq2 with x0, x2, x4, x6, x8, x10, λ x11 x12 : ο . x12 = x7 x10 leaving 2 subgoals.

The subproof is completed by applying H3.

Claim L4: x10 ∈ x1

Apply L2 with λ x11 x12 . x10 ∈ x11.

The subproof is completed by applying H3.

Apply H0 with λ x11 x12 . decode_p (ap x12 3) x10 = x7 x10.

Let x11 of type ο → ο → ο be given.

Apply pack_c_r_p_p_3_eq2 with x1, x3, x5, x7, x9, x10, λ x12 x13 : ο . x11 x13 x12.

The subproof is completed by applying L4.

Let x10 of type ι be given.

Assume H3: x10 ∈ x0.

Apply pack_c_r_p_p_4_eq2 with x0, x2, x4, x6, x8, x10, λ x11 x12 : ο . x12 = x9 x10 leaving 2 subgoals.

The subproof is completed by applying H3.

Claim L4: x10 ∈ x1

Apply L2 with λ x11 x12 . x10 ∈ x11.

The subproof is completed by applying H3.

Apply H0 with λ x11 x12 . decode_p (ap x12 4) x10 = x9 x10.

Let x11 of type ο → ο → ο be given.

Apply pack_c_r_p_p_4_eq2 with x1, x3, x5, x7, x9, x10, λ x12 x13 : ο . x11 x13 x12.

The subproof is completed by applying L4.

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Proofgold Proof