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Proofgold Proof

pf
Let x0 of type ι be given.
Assume H0: SNo x0.
Apply xm with SNoLe 0 x0, SNoLev (abs_SNo x0) = SNoLev x0 leaving 2 subgoals.
Assume H1: SNoLe 0 x0.
Apply nonneg_abs_SNo with x0, λ x1 x2 . SNoLev x2 = SNoLev x0 leaving 2 subgoals.
The subproof is completed by applying H1.
Let x1 of type ιιο be given.
Assume H2: x1 (SNoLev x0) (SNoLev x0).
The subproof is completed by applying H2.
Assume H1: not (SNoLe 0 x0).
Apply not_nonneg_abs_SNo with x0, λ x1 x2 . SNoLev x2 = SNoLev x0 leaving 2 subgoals.
The subproof is completed by applying H1.
Apply minus_SNo_Lev with x0.
The subproof is completed by applying H0.