Let x0 of type ι be given.
Apply SNoLtLe_or with
x0,
0,
SNoLe 0 (abs_SNo 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 . SNoLe 0 x2 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply SNoLtLe with
0,
minus_SNo x0.
Apply minus_SNo_Lt_contra2 with
x0,
0 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying SNo_0.
Apply minus_SNo_0 with
λ x1 x2 . SNoLt x0 x2.
The subproof is completed by applying H1.
Apply nonneg_abs_SNo with
x0,
λ x1 x2 . SNoLe 0 x2 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H1.