Let x0 of type ι be given.
Let x1 of type ι → ι be given.
Let x2 of type ι → ι be given.
Let x3 of type ι be given.
Assume H0: ∀ x4 . x4 ∈ x0 ⟶ x1 x4 = x2 x4.
Claim L1:
lam x0 x1 = lam x0 x2Apply encode_u_ext with
x0,
x1,
x2.
The subproof is completed by applying H0.
Apply L1 with
λ x4 x5 . lam 3 (λ x6 . If_i (x6 = 0) x0 (If_i (x6 = 1) (lam x0 x1) x3)) = lam 3 (λ x6 . If_i (x6 = 0) x0 (If_i (x6 = 1) x4 x3)).
Let x4 of type ι → ι → ο be given.
Assume H2:
x4 (lam 3 (λ x5 . If_i (x5 = 0) x0 (If_i (x5 = 1) (lam x0 x1) x3))) (lam 3 (λ x5 . If_i (x5 = 0) x0 (If_i (x5 = 1) (lam x0 x1) x3))).
The subproof is completed by applying H2.