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Proofgold Proof

pf
Let x0 of type ι be given.
Apply nat_ind with λ x1 . x0add_nat x0 x1 leaving 2 subgoals.
Apply add_nat_0R with x0, λ x1 x2 . x0x2.
The subproof is completed by applying Subq_ref with x0.
Let x1 of type ι be given.
Assume H0: nat_p x1.
Assume H1: x0add_nat x0 x1.
Apply add_nat_SR with x0, x1, λ x2 x3 . x0x3 leaving 2 subgoals.
The subproof is completed by applying H0.
Apply Subq_tra with x0, add_nat x0 x1, ordsucc (add_nat x0 x1) leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying ordsuccI1 with add_nat x0 x1.