Let x0 of type ι be given.
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.
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.