Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Apply beta with
3,
λ x3 . If_i (x3 = 0) x0 (If_i (x3 = 1) x1 x2),
0,
λ x3 x4 . x4 = x0 leaving 2 subgoals.
The subproof is completed by applying In_0_3.
Apply If_i_1 with
0 = 0,
x0,
If_i (0 = 1) x1 x2.
Let x3 of type ι → ι → ο be given.
Assume H0: x3 0 0.
The subproof is completed by applying H0.