Let x0 of type ι be given.
Let x1 of type ι be given.
Assume H1: x1 ∈ x0.
Apply nat_ind with
λ x2 . add_nat x1 x2 ∈ add_nat x0 x2 leaving 2 subgoals.
Apply add_nat_0R with
x1,
λ x2 x3 . x3 ∈ add_nat x0 0.
Apply add_nat_0R with
x0,
λ x2 x3 . x1 ∈ x3.
The subproof is completed by applying H1.
Let x2 of type ι be given.
Apply add_nat_SR with
x1,
x2,
λ x3 x4 . x4 ∈ add_nat x0 (ordsucc x2) leaving 2 subgoals.
The subproof is completed by applying H2.
Apply add_nat_SR with
x0,
x2,
λ x3 x4 . ordsucc (add_nat x1 x2) ∈ x4 leaving 2 subgoals.
The subproof is completed by applying H2.
Apply nat_ordsucc_in_ordsucc with
add_nat x0 x2,
add_nat x1 x2 leaving 2 subgoals.
Apply add_nat_p with
x0,
x2 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H2.
The subproof is completed by applying H3.