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