Let x0 of type ι be given.
Let x1 of type (ι → ο) → ο be given.
Let x2 of type (ι → ο) → ο be given.
Let x3 of type ι → ι be given.
Let x4 of type ι → ι be given.
Let x5 of type ι → ι → ο be given.
Let x6 of type ι → ι → ο be given.
Assume H0:
∀ x7 : ι → ο . (∀ x8 . x7 x8 ⟶ x8 ∈ x0) ⟶ iff (x1 x7) (x2 x7).
Assume H1: ∀ x7 . x7 ∈ x0 ⟶ x3 x7 = x4 x7.
Assume H2:
∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ iff (x5 x7 x8) (x6 x7 x8).
Apply L3 with
λ x7 x8 . lam 4 (λ x9 . If_i (x9 = 0) x0 (If_i (x9 = 1) (encode_c x0 x1) (If_i (x9 = 2) (lam x0 x3) (encode_r x0 x5)))) = lam 4 (λ x9 . If_i (x9 = 0) x0 (If_i (x9 = 1) x7 (If_i (x9 = 2) (lam x0 x4) (encode_r x0 x6)))).
Claim L4:
lam x0 x3 = lam x0 x4Apply encode_u_ext with
x0,
x3,
x4.
The subproof is completed by applying H1.
Apply L4 with
λ x7 x8 . lam 4 (λ x9 . If_i (x9 = 0) x0 (If_i (x9 = 1) (encode_c x0 x1) (If_i (x9 = 2) (lam x0 x3) (encode_r x0 x5)))) = lam 4 (λ x9 . If_i (x9 = 0) x0 (If_i (x9 = 1) (encode_c x0 x1) (If_i (x9 = 2) x7 (encode_r x0 x6)))).
Apply encode_r_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_c x0 x1) (If_i (x9 = 2) (lam x0 x3) (encode_r x0 x5)))) = lam 4 (λ x9 . If_i (x9 = 0) x0 (If_i (x9 = 1) (encode_c x0 x1) (If_i (x9 = 2) (lam x0 x3) x7))).
Let x7 of type ι → ι → ο be given.
The subproof is completed by applying H6.