Let x0 of type ι be given.
Apply nat_ind with
λ x1 . SNoLt 0 (exp_SNo_nat x0 x1) leaving 2 subgoals.
Apply exp_SNo_nat_0 with
x0,
λ x1 x2 . SNoLt 0 x2 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying SNoLt_0_1.
Let x1 of type ι be given.
Apply exp_SNo_nat_S with
x0,
x1,
λ x2 x3 . SNoLt 0 x3 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H2.
Apply mul_SNo_zeroR with
x0,
λ x2 x3 . SNoLt x2 (mul_SNo x0 (exp_SNo_nat x0 x1)) leaving 2 subgoals.
The subproof is completed by applying H0.
Apply pos_mul_SNo_Lt with
x0,
0,
exp_SNo_nat x0 x1 leaving 5 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying SNo_0.
Apply SNo_exp_SNo_nat with
x0,
x1 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H2.
The subproof is completed by applying H3.