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