Let x0 of type ι be given.
Let x1 of type ι be given.
Apply beta with
2,
λ x2 . If_i (x2 = 0) x0 x1,
0,
λ x2 x3 . x3 = x0 leaving 2 subgoals.
The subproof is completed by applying In_0_2.
Apply If_i_1 with
0 = 0,
x0,
x1.
Let x2 of type ι → ι → ο be given.
Assume H0: x2 0 0.
The subproof is completed by applying H0.