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.
Let x4 of type ι be given.
Assume H0: x0 ∈ x4.
Assume H1: x1 ∈ x4.
Assume H2: x2 ∈ x4.
Assume H3: x3 ∈ x4.
Apply tuple_4_eta with
x0,
x1,
x2,
x3,
λ x5 x6 . x5 ∈ setexp x4 4.
Apply lam_Pi with
4,
λ x5 . x4,
λ x5 . ap (lam 4 (λ x6 . If_i (x6 = 0) x0 (If_i (x6 = 1) x1 (If_i (x6 = 2) x2 x3)))) x5.
Let x5 of type ι be given.
Assume H4: x5 ∈ 4.
Apply cases_4 with
x5,
λ x6 . ap (lam 4 (λ x7 . If_i (x7 = 0) x0 (If_i (x7 = 1) x1 (If_i (x7 = 2) x2 x3)))) x6 ∈ x4 leaving 5 subgoals.
The subproof is completed by applying H4.
Apply tuple_4_0_eq with
x0,
x1,
x2,
x3,
λ x6 x7 . x7 ∈ x4.
The subproof is completed by applying H0.
Apply tuple_4_1_eq with
x0,
x1,
x2,
x3,
λ x6 x7 . x7 ∈ x4.
The subproof is completed by applying H1.
Apply tuple_4_2_eq with
x0,
x1,
x2,
x3,
λ x6 x7 . x7 ∈ x4.
The subproof is completed by applying H2.
Apply tuple_4_3_eq with
x0,
x1,
x2,
x3,
λ x6 x7 . x7 ∈ x4.
The subproof is completed by applying H3.