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Proofgold Proof

pf
Let x0 of type ι be given.
Assume H0: SNo x0.
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.
Assume H1: SNoLt x0 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.
Assume H1: SNoLe 0 x0.
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.