Let x0 of type ι be given.
Let x1 of type ι be given.
Apply nat_ind with
λ x2 . add_nat x0 x2 = add_nat x1 x2 ⟶ x0 = x1 leaving 2 subgoals.
Apply add_nat_0R with
x0,
λ x2 x3 . x3 = add_nat x1 0 ⟶ x0 = x1.
Apply add_nat_0R with
x1,
λ x2 x3 . x0 = x3 ⟶ x0 = x1.
Assume H2: x0 = x1.
The subproof is completed by applying H2.
Let x2 of type ι be given.
Apply add_nat_SR with
x0,
x2,
λ x3 x4 . x4 = add_nat x1 (ordsucc x2) ⟶ x0 = x1 leaving 2 subgoals.
The subproof is completed by applying H2.
Apply add_nat_SR with
x1,
x2,
λ x3 x4 . ordsucc (add_nat x0 x2) = x4 ⟶ x0 = x1 leaving 2 subgoals.
The subproof is completed by applying H2.
Apply H3.
Apply ordsucc_inj with
add_nat x0 x2,
add_nat x1 x2.
The subproof is completed by applying H4.