Let x0 of type ι be given.
Apply nat_ind with
λ x1 . add_nat x0 x1 = 0 ⟶ and (x0 = 0) (x1 = 0) leaving 2 subgoals.
Apply add_nat_0R with
x0,
λ x1 x2 . x2 = 0 ⟶ and (x0 = 0) (0 = 0).
Assume H0: x0 = 0.
Apply andI with
x0 = 0,
0 = 0 leaving 2 subgoals.
The subproof is completed by applying H0.
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.
Assume H1:
add_nat x0 x1 = 0 ⟶ and (x0 = 0) (x1 = 0).
Apply add_nat_SR with
x0,
x1,
λ x2 x3 . x3 = 0 ⟶ and (x0 = 0) (ordsucc x1 = 0) leaving 2 subgoals.
The subproof is completed by applying H0.
Apply FalseE with
and (x0 = 0) (ordsucc x1 = 0).
Apply neq_ordsucc_0 with
add_nat x0 x1.
The subproof is completed by applying H2.