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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 (mul_SNo x0 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 SNoLtLe with 0, mul_SNo x0 x0.
Apply mul_SNo_zeroR with x0, λ x1 x2 . SNoLt x1 (mul_SNo x0 x0) leaving 2 subgoals.
The subproof is completed by applying H0.
Apply neg_mul_SNo_Lt with x0, 0, x0 leaving 5 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying SNo_0.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Assume H1: SNoLe 0 x0.
Apply mul_SNo_zeroR with x0, λ x1 x2 . SNoLe x1 (mul_SNo x0 x0) leaving 2 subgoals.
The subproof is completed by applying H0.
Apply nonneg_mul_SNo_Le with x0, 0, x0 leaving 5 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying SNo_0.
The subproof is completed by applying H0.
The subproof is completed by applying H1.