Let x0 of type ι be given.
Apply nat_ind with
λ x1 . add_nat x0 x1 = x1 ⟶ x0 = 0 leaving 2 subgoals.
Apply add_nat_0R with
x0,
λ x1 x2 . x2 = 0 ⟶ x0 = 0.
Assume H0: x0 = 0.
The subproof is completed by applying H0.
Let x1 of type ι be given.
Assume H1:
add_nat x0 x1 = x1 ⟶ x0 = 0.
Apply add_nat_SR with
x0,
x1,
λ x2 x3 . x3 = ordsucc x1 ⟶ x0 = 0 leaving 2 subgoals.
The subproof is completed by applying H0.
Apply H1.
Apply ordsucc_inj with
add_nat x0 x1,
x1.
The subproof is completed by applying H2.