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 → ι be given.

Let x4 of type ι → ι → ι be given.

Let x5 of type ι be given.

Let x6 of type ι be given.

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

Assume H1: ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x3 x7 x8 = x4 x7 x8.

Claim L2: encode_b x0 x1 = encode_b x0 x2

Apply encode_b_ext with x0, x1, x2.

The subproof is completed by applying H0.

Apply L2 with λ x7 x8 . lam 5 (λ x9 . If_i (x9 = 0) x0 (If_i (x9 = 1) (encode_b x0 x1) (If_i (x9 = 2) (encode_b x0 x3) (If_i (x9 = 3) x5 x6)))) = lam 5 (λ x9 . If_i (x9 = 0) x0 (If_i (x9 = 1) x7 (If_i (x9 = 2) (encode_b x0 x4) (If_i (x9 = 3) x5 x6)))).

Claim L3: encode_b x0 x3 = encode_b x0 x4

Apply encode_b_ext with x0, x3, x4.

The subproof is completed by applying H1.

Apply L3 with λ x7 x8 . lam 5 (λ x9 . If_i (x9 = 0) x0 (If_i (x9 = 1) (encode_b x0 x1) (If_i (x9 = 2) (encode_b x0 x3) (If_i (x9 = 3) x5 x6)))) = lam 5 (λ x9 . If_i (x9 = 0) x0 (If_i (x9 = 1) (encode_b x0 x1) (If_i (x9 = 2) x7 (If_i (x9 = 3) x5 x6)))).

Let x7 of type ι → ι → ο be given.

Assume H4: x7 (lam 5 (λ x8 . If_i (x8 = 0) x0 (If_i (x8 = 1) (encode_b x0 x1) (If_i (x8 = 2) (encode_b x0 x3) (If_i (x8 = 3) x5 x6))))) (lam 5 (λ x8 . If_i (x8 = 0) x0 (If_i (x8 = 1) (encode_b x0 x1) (If_i (x8 = 2) (encode_b x0 x3) (If_i (x8 = 3) x5 x6))))).

The subproof is completed by applying H4.

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Proofgold Proof