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.
Apply beta with
4,
λ x4 . If_i (x4 = 0) x0 (If_i (x4 = 1) x1 (If_i (x4 = 2) x2 x3)),
0,
λ x4 x5 . x5 = x0 leaving 2 subgoals.
The subproof is completed by applying In_0_4.
Apply If_i_1 with
0 = 0,
x0,
If_i (0 = 1) x1 (If_i (0 = 2) x2 x3).
Let x4 of type ι → ι → ο be given.
Assume H0: x4 0 0.
The subproof is completed by applying H0.