Let x0 of type ι be given.
Let x1 of type ι → ο be given.
Let x2 of type ι → ο be given.
Assume H0:
∀ x3 . x3 ∈ x0 ⟶ iff (x1 x3) (x2 x3).
Claim L1:
Sep x0 x1 = Sep x0 x2Apply encode_p_ext with
x0,
x1,
x2.
The subproof is completed by applying H0.
Apply L1 with
λ x3 x4 . lam 2 (λ x5 . If_i (x5 = 0) x0 x4) = lam 2 (λ x5 . If_i (x5 = 0) x0 (Sep x0 x2)).
Let x3 of type ι → ι → ο be given.
Assume H2:
x3 (lam 2 (λ x4 . If_i (x4 = 0) x0 (Sep x0 x2))) (lam 2 (λ x4 . If_i (x4 = 0) x0 (Sep x0 x2))).
The subproof is completed by applying H2.