Let x0 of type ι be given.
Apply nat_ind with
λ x1 . nat_p (exp_SNo_nat x0 x1) leaving 2 subgoals.
Apply exp_SNo_nat_0 with
x0,
λ x1 x2 . nat_p x2 leaving 2 subgoals.
Apply nat_p_SNo with
x0.
The subproof is completed by applying H0.
The subproof is completed by applying nat_1.
Let x1 of type ι be given.
Apply exp_SNo_nat_S with
x0,
x1,
λ x2 x3 . nat_p x3 leaving 3 subgoals.
Apply nat_p_SNo with
x0.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply mul_nat_mul_SNo with
x0,
exp_SNo_nat x0 x1,
λ x2 x3 . nat_p x2 leaving 3 subgoals.
Apply nat_p_omega with
x0.
The subproof is completed by applying H0.
Apply nat_p_omega with
exp_SNo_nat x0 x1.
The subproof is completed by applying H2.
Apply mul_nat_p with
x0,
exp_SNo_nat x0 x1 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H2.