Let x0 of type ι be given.
Let x1 of type ι → ι be given.
Let x2 of type ι → ι → ο be given.
Let x3 of type ι be given.
Assume H0: x3 ∈ x0.
Let x4 of type ι be given.
Assume H1: x4 ∈ x1 x3.
Assume H2: x2 x3 x4.
Apply SepI with
lam x0 (λ x5 . x1 x5),
λ x5 . x2 (ap x5 0) (ap x5 1),
lam 2 (λ x5 . If_i (x5 = 0) x3 x4) leaving 2 subgoals.
Apply tuple_2_Sigma with
x0,
λ x5 . x1 x5,
x3,
x4 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply tuple_2_0_eq with
x3,
x4,
λ x5 x6 . x2 x6 (ap (lam 2 (λ x7 . If_i (x7 = 0) x3 x4)) 1).
Apply tuple_2_1_eq with
x3,
x4,
λ x5 x6 . x2 x3 x6.
The subproof is completed by applying H2.