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Proofgold Proof

pf
Let x0 of type ι be given.
Apply nat_ind with λ x1 . add_nat x0 x1 = x1x0 = 0 leaving 2 subgoals.
Apply add_nat_0R with x0, λ x1 x2 . x2 = 0x0 = 0.
Assume H0: x0 = 0.
The subproof is completed by applying H0.
Let x1 of type ι be given.
Assume H0: nat_p x1.
Assume H1: add_nat x0 x1 = x1x0 = 0.
Apply add_nat_SR with x0, x1, λ x2 x3 . x3 = ordsucc x1x0 = 0 leaving 2 subgoals.
The subproof is completed by applying H0.
Assume H2: ordsucc (add_nat x0 x1) = ordsucc x1.
Apply H1.
Apply ordsucc_inj with add_nat x0 x1, x1.
The subproof is completed by applying H2.