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Proofgold Proof

pf
Let x0 of type ι be given.
Assume H0: nat_p x0.
Apply nat_ind with λ x1 . nat_p (exp_nat x0 x1) leaving 2 subgoals.
Apply nat_primrec_0 with 1, λ x1 x2 . mul_nat x0 x2, λ x1 x2 . nat_p x2.
The subproof is completed by applying nat_1.
Let x1 of type ι be given.
Assume H1: nat_p x1.
Assume H2: nat_p (exp_nat x0 x1).
Apply nat_primrec_S with 1, λ x2 x3 . mul_nat x0 x3, x1, λ x2 x3 . nat_p x3 leaving 2 subgoals.
The subproof is completed by applying H1.
Apply mul_nat_p with x0, exp_nat x0 x1 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H2.