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.
Let x6 of type ι be given.
Assume H0:
∀ x7 . x7 ∈ x0 ⟶ iff (x1 x7) (x2 x7).
Assume H1:
∀ x7 . x7 ∈ x0 ⟶ iff (x3 x7) (x4 x7).
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 x7 x8 . lam 5 (λ x9 . If_i (x9 = 0) x0 (If_i (x9 = 1) (Sep x0 x1) (If_i (x9 = 2) (Sep x0 x3) (If_i (x9 = 3) x5 x6)))) = lam 5 (λ x9 . If_i (x9 = 0) x0 (If_i (x9 = 1) x7 (If_i (x9 = 2) (Sep x0 x4) (If_i (x9 = 3) x5 x6)))).
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 x7 x8 . lam 5 (λ x9 . If_i (x9 = 0) x0 (If_i (x9 = 1) (Sep x0 x1) (If_i (x9 = 2) (Sep x0 x3) (If_i (x9 = 3) x5 x6)))) = lam 5 (λ x9 . If_i (x9 = 0) x0 (If_i (x9 = 1) (Sep x0 x1) (If_i (x9 = 2) x7 (If_i (x9 = 3) x5 x6)))).
Let x7 of type ι → ι → ο be given.
The subproof is completed by applying H4.