Let x0 of type ι be given.
Apply add_SNo_1_1_2 with
λ x1 x2 . mul_SNo x1 x0 = add_SNo x0 x0.
Apply mul_SNo_distrR with
1,
1,
x0,
λ x1 x2 . x2 = add_SNo x0 x0 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 . add_SNo x2 x2 = add_SNo x0 x0 leaving 2 subgoals.
The subproof is completed by applying H0.
Let x1 of type ι → ι → ο be given.
The subproof is completed by applying H1.