Let x0 of type ι be given.
Apply SNoLtLe_or with
x0,
0,
mul_SNo (abs_SNo x0) (abs_SNo x0) = mul_SNo x0 x0 leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying SNo_0.
Apply neg_abs_SNo with
x0,
λ x1 x2 . mul_SNo x2 x2 = mul_SNo x0 x0 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply mul_SNo_minus_distrL with
x0,
minus_SNo x0,
λ x1 x2 . x2 = mul_SNo x0 x0 leaving 3 subgoals.
The subproof is completed by applying H0.
Apply SNo_minus_SNo with
x0.
The subproof is completed by applying H0.
Apply mul_SNo_minus_distrR with
x0,
x0,
λ x1 x2 . minus_SNo x2 = mul_SNo x0 x0 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H0.
Apply minus_SNo_invol with
mul_SNo x0 x0.
Apply SNo_mul_SNo with
x0,
x0 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H0.
Apply nonneg_abs_SNo with
x0,
λ x1 x2 . mul_SNo x2 x2 = mul_SNo x0 x0 leaving 2 subgoals.
The subproof is completed by applying H1.
Let x1 of type ι → ι → ο be given.
The subproof is completed by applying H2.