Let x0 of type ι be given.
Apply nat_ind with
λ x1 . x0 ∈ ordsucc (add_nat x0 x1) leaving 2 subgoals.
Apply add_nat_0R with
x0,
λ x1 x2 . x0 ∈ ordsucc x2.
The subproof is completed by applying ordsuccI2 with x0.
Let x1 of type ι be given.
Apply add_nat_SR with
x0,
x1,
λ x2 x3 . x0 ∈ ordsucc x3 leaving 2 subgoals.
The subproof is completed by applying H0.
Apply ordsuccI1 with
ordsucc (add_nat x0 x1),
x0.
The subproof is completed by applying H1.