Let x0 of type ι be given.
Apply add_SNo_1_1_2 with
λ x1 x2 . SNoLt (add_SNo x0 1) (mul_SNo x1 x0).
Apply mul_SNo_distrR with
1,
1,
x0,
λ x1 x2 . SNoLt (add_SNo x0 1) x2 leaving 4 subgoals.
The subproof is completed by applying SNo_1.
The subproof is completed by applying SNo_1.
The subproof is completed by applying H0.
Apply mul_SNo_oneL with
x0,
λ x1 x2 . SNoLt (add_SNo x0 1) (add_SNo x2 x2) leaving 2 subgoals.
The subproof is completed by applying H0.
Apply add_SNo_Lt2 with
x0,
1,
x0 leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying SNo_1.
The subproof is completed by applying H0.
The subproof is completed by applying H1.