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Proofgold Proof

pf
Let x0 of type ι be given.
Assume H0: SNo x0.
Assume H1: SNoLt x0 0.
Claim L2: SNoLt 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 recip_SNo_negcase with x0, λ x1 x2 . SNoLt x2 0 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply minus_SNo_Lt_contra1 with 0, recip_SNo_pos (minus_SNo x0) leaving 3 subgoals.
The subproof is completed by applying SNo_0.
Apply SNo_recip_SNo_pos with minus_SNo x0 leaving 2 subgoals.
Apply SNo_minus_SNo with x0.
The subproof is completed by applying H0.
The subproof is completed by applying L2.
Apply minus_SNo_0 with λ x1 x2 . SNoLt x2 (recip_SNo_pos (minus_SNo x0)).
Apply recip_SNo_pos_pos with minus_SNo x0 leaving 2 subgoals.
Apply SNo_minus_SNo with x0.
The subproof is completed by applying H0.
The subproof is completed by applying L2.