Let x0 of type ι be given.
Let x1 of type ι → ο be given.
Let x2 of type ι → ο be given.
Let x3 of type ι → ο be given.
Let x4 of type ι → ο be given.
Let x5 of type ι be given.
Assume H0:
∀ x6 . x6 ∈ x0 ⟶ iff (x1 x6) (x2 x6).
Assume H1:
∀ x6 . x6 ∈ x0 ⟶ iff (x3 x6) (x4 x6).
Claim L2:
Sep x0 x1 = Sep x0 x2Apply encode_p_ext with
x0,
x1,
x2.
The subproof is completed by applying H0.
Apply L2 with
λ x6 x7 . lam 4 (λ x8 . If_i (x8 = 0) x0 (If_i (x8 = 1) (Sep x0 x1) (If_i (x8 = 2) (Sep x0 x3) x5))) = lam 4 (λ x8 . If_i (x8 = 0) x0 (If_i (x8 = 1) x6 (If_i (x8 = 2) (Sep x0 x4) x5))).
Claim L3:
Sep x0 x3 = Sep x0 x4Apply encode_p_ext with
x0,
x3,
x4.
The subproof is completed by applying H1.
Apply L3 with
λ x6 x7 . lam 4 (λ x8 . If_i (x8 = 0) x0 (If_i (x8 = 1) (Sep x0 x1) (If_i (x8 = 2) (Sep x0 x3) x5))) = lam 4 (λ x8 . If_i (x8 = 0) x0 (If_i (x8 = 1) (Sep x0 x1) (If_i (x8 = 2) x6 x5))).
Let x6 of type ι → ι → ο be given.
The subproof is completed by applying H4.