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Proofgold Proof

pf
Let x0 of type ι be given.
Assume H0: SNo x0.
Apply SNoLt_trichotomy_or_impred with x0, 0, SNo (recip_SNo x0) leaving 5 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying SNo_0.
Assume H1: SNoLt x0 0.
Apply recip_SNo_negcase with x0, λ x1 x2 . SNo x2 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply SNo_minus_SNo with recip_SNo_pos (minus_SNo x0).
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.
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: x0 = 0.
Apply H1 with λ x1 x2 . SNo (recip_SNo x2).
Apply recip_SNo_0 with λ x1 x2 . SNo x2.
The subproof is completed by applying SNo_0.
Assume H1: SNoLt 0 x0.
Apply recip_SNo_poscase with x0, λ x1 x2 . SNo x2 leaving 2 subgoals.
The subproof is completed by applying H1.
Apply SNo_recip_SNo_pos with x0 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.