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Proofgold Proof

pf
Apply nat_ind with λ x0 . mul_nat 0 x0 = 0 leaving 2 subgoals.
The subproof is completed by applying mul_nat_0R with 0.
Let x0 of type ι be given.
Assume H0: nat_p x0.
Assume H1: mul_nat 0 x0 = 0.
Apply mul_nat_SR with 0, x0, λ x1 x2 . x2 = 0 leaving 2 subgoals.
The subproof is completed by applying H0.
Apply H1 with λ x1 x2 . add_nat 0 x2 = 0.
The subproof is completed by applying add_nat_0R with 0.