Let x0 of type ι be given.
Let x1 of type ι → ι → ο be given.
Let x2 of type ι → ι → ο be given.
Assume H0:
∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ iff (x1 x3 x4) (x2 x3 x4).
Apply encode_r_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 (encode_r x0 x2)).
Let x3 of type ι → ι → ο be given.
The subproof is completed by applying H2.