Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι → ι → ι be given.
Let x3 of type ι be given.
Assume H0: x3 ∈ x1.
Apply beta with
x1,
λ x4 . If_i (x4 = x3) x0 (x2 (ordsucc x3) x4),
x3,
λ x4 x5 . x5 = x0 leaving 2 subgoals.
The subproof is completed by applying H0.
Apply If_i_1 with
x3 = x3,
x0,
x2 (ordsucc x3) x3.
Let x4 of type ι → ι → ο be given.
Assume H1: x4 x3 x3.
The subproof is completed by applying H1.