Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Assume H0: x0 ∈ x2.
Assume H1: x1 ∈ x2.
Apply tuple_2_eta with
x0,
x1,
λ x3 x4 . x3 ∈ setexp x2 2.
Apply lam_Pi with
2,
λ x3 . x2,
λ x3 . ap (lam 2 (λ x4 . If_i (x4 = 0) x0 x1)) x3.
Let x3 of type ι be given.
Assume H2: x3 ∈ 2.
Apply cases_2 with
x3,
λ x4 . ap (lam 2 (λ x5 . If_i (x5 = 0) x0 x1)) x4 ∈ x2 leaving 3 subgoals.
The subproof is completed by applying H2.
Apply tuple_2_0_eq with
x0,
x1,
λ x4 x5 . x5 ∈ x2.
The subproof is completed by applying H0.
Apply tuple_2_1_eq with
x0,
x1,
λ x4 x5 . x5 ∈ x2.
The subproof is completed by applying H1.