Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Let x3 of type ι be given.
Assume H0: x0 ∈ x3.
Assume H1: x1 ∈ x3.
Assume H2: x2 ∈ x3.
Apply tuple_3_eta with
x0,
x1,
x2,
λ x4 x5 . x4 ∈ setexp x3 3.
Apply lam_Pi with
3,
λ x4 . x3,
λ x4 . ap (lam 3 (λ x5 . If_i (x5 = 0) x0 (If_i (x5 = 1) x1 x2))) x4.
Let x4 of type ι be given.
Assume H3: x4 ∈ 3.
Apply cases_3 with
x4,
λ x5 . ap (lam 3 (λ x6 . If_i (x6 = 0) x0 (If_i (x6 = 1) x1 x2))) x5 ∈ x3 leaving 4 subgoals.
The subproof is completed by applying H3.
Apply tuple_3_0_eq with
x0,
x1,
x2,
λ x5 x6 . x6 ∈ x3.
The subproof is completed by applying H0.
Apply tuple_3_1_eq with
x0,
x1,
x2,
λ x5 x6 . x6 ∈ x3.
The subproof is completed by applying H1.
Apply tuple_3_2_eq with
x0,
x1,
x2,
λ x5 x6 . x6 ∈ x3.
The subproof is completed by applying H2.