Let x0 of type ι be given.
Apply nat_ind with
λ x1 . or (x1 = 0) (x0 ∈ add_nat x0 x1) leaving 2 subgoals.
Apply orIL with
0 = 0,
x0 ∈ add_nat x0 0.
Let x1 of type ι → ι → ο be given.
Assume H1: x1 0 0.
The subproof is completed by applying H1.
Let x1 of type ι be given.
Apply orIR with
ordsucc x1 = 0,
x0 ∈ add_nat x0 (ordsucc x1).
Apply add_nat_SR with
x0,
x1,
λ x2 x3 . x0 ∈ x3 leaving 2 subgoals.
The subproof is completed by applying H1.
Apply H2 with
x0 ∈ ordsucc (add_nat x0 x1) leaving 2 subgoals.
Assume H3: x1 = 0.
Apply H3 with
λ x2 x3 . x0 ∈ ordsucc (add_nat x0 x3).
Apply add_nat_0R with
x0,
λ x2 x3 . x0 ∈ ordsucc x3.
The subproof is completed by applying ordsuccI2 with x0.
Apply ordsuccI1 with
add_nat x0 x1,
x0.
The subproof is completed by applying H3.