Let x0 of type ι be given.

Let x1 of type ι → ι → ι be given.

Let x2 of type ι → ι → ι be given.

Let x3 of type ι → ι → ο be given.

Let x4 of type ι → ι → ο be given.

Let x5 of type ι → ο be given.

Let x6 of type ι → ο be given.

Assume H0: ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x1 x7 x8 = x2 x7 x8.

Assume H1: ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ iff (x3 x7 x8) (x4 x7 x8).

Assume H2: ∀ x7 . x7 ∈ x0 ⟶ iff (x5 x7) (x6 x7).

Claim L3: ...

...

Apply L3 with λ x7 x8 . lam 4 (λ x9 . If_i (x9 = 0) x0 (If_i (x9 = 1) (encode_b x0 x1) (If_i (x9 = 2) (encode_r x0 x3) (Sep x0 x5)))) = lam 4 (λ x9 . If_i (x9 = 0) x0 (If_i (x9 = 1) x7 (If_i (x9 = 2) (encode_r x0 x4) (Sep x0 x6)))).

Claim L4: encode_r x0 x3 = encode_r x0 x4

Apply encode_r_ext with x0, x3, x4.

...

Apply L4 with λ x7 x8 . lam 4 (λ x9 . If_i (x9 = 0) x0 (If_i (x9 = 1) (encode_b x0 x1) (If_i (x9 = 2) (encode_r x0 x3) (Sep x0 x5)))) = lam 4 (λ x9 . If_i (x9 = 0) x0 (If_i (x9 = 1) (encode_b x0 x1) (If_i (x9 = 2) x7 (Sep x0 x6)))).

Claim L5: Sep x0 x5 = Sep x0 x6

Apply encode_p_ext with x0, x5, x6.

The subproof is completed by applying H2.

Apply L5 with λ x7 x8 . lam 4 (λ x9 . If_i (x9 = 0) x0 (If_i (x9 = 1) (encode_b x0 x1) (If_i (x9 = 2) (encode_r x0 x3) (Sep x0 x5)))) = lam 4 (λ x9 . If_i (x9 = 0) x0 (If_i (x9 = 1) (encode_b x0 x1) (If_i (x9 = 2) (encode_r x0 x3) x7))).

Let x7 of type ι → ι → ο be given.

Assume H6: x7 (lam 4 (λ x8 . If_i (x8 = 0) x0 (If_i (x8 = 1) (encode_b x0 x1) (If_i (x8 = 2) (encode_r x0 x3) (Sep x0 x5))))) (lam 4 (λ x8 . If_i (x8 = 0) x0 (If_i (x8 = 1) (encode_b x0 x1) (If_i (x8 = 2) (encode_r x0 x3) (Sep x0 x5))))).

The subproof is completed by applying H6.

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Proofgold Proof