Let x0 of type ι be given.
Apply add_nat_add_SNo with
x0,
1,
λ x1 x2 . x1 = ordsucc x0 leaving 3 subgoals.
The subproof is completed by applying H0.
Apply nat_p_omega with
1.
The subproof is completed by applying nat_1.
Apply add_nat_SR with
x0,
0,
λ x1 x2 . x2 = ordsucc x0 leaving 2 subgoals.
The subproof is completed by applying nat_0.
Apply add_nat_0R with
x0,
λ x1 x2 . ordsucc x2 = ordsucc x0.
Let x2 of type ι → ι → ο be given.
Assume H1: x2 y1 y1.
The subproof is completed by applying H1.