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Proofgold Proof

pf
Let x0 of type ι be given.
Assume H0: x0real.
Claim L1: SNo x0
Apply real_SNo with x0.
The subproof is completed by applying H0.
Apply SNoLt_trichotomy_or_impred with x0, 0, recip_SNo x0real leaving 5 subgoals.
The subproof is completed by applying L1.
The subproof is completed by applying SNo_0.
Assume H2: SNoLt x0 0.
Apply recip_SNo_negcase with x0, λ x1 x2 . x2real leaving 3 subgoals.
The subproof is completed by applying L1.
The subproof is completed by applying H2.
Apply real_minus_SNo with recip_SNo_pos (minus_SNo x0).
Apply real_recip_SNo_pos with minus_SNo x0 leaving 2 subgoals.
Apply real_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 L1.
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 H2.
Assume H2: x0 = 0.
Apply H2 with λ x1 x2 . recip_SNo x2real.
Apply recip_SNo_0 with λ x1 x2 . x2real.
The subproof is completed by applying real_0.
Assume H2: SNoLt 0 x0.
Apply recip_SNo_poscase with x0, λ x1 x2 . x2real leaving 2 subgoals.
The subproof is completed by applying H2.
Apply real_recip_SNo_pos with x0 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H2.