Let x be given.
Assume Hx.
Apply SNoLt_trichotomy_or_impred x 0 Hx SNo_0 to the current goal.
Assume H1: x < 0.
rewrite the current goal using recip_SNo_negcase x Hx H1 (from left to right).
We will prove SNo (- recip_SNo_pos (- x)).
Apply SNo_minus_SNo to the current goal.
We will prove SNo (recip_SNo_pos (- x)).
Apply SNo_recip_SNo_pos to the current goal.
An exact proof term for the current goal is SNo_minus_SNo x Hx.
We will prove 0 < - x.
Apply minus_SNo_Lt_contra2 x 0 Hx SNo_0 to the current goal.
We will prove x < - 0.
rewrite the current goal using minus_SNo_0 (from left to right).
We will prove x < 0.
An exact proof term for the current goal is H1.
Assume H1: x = 0.
rewrite the current goal using H1 (from left to right).
rewrite the current goal using recip_SNo_0 (from left to right).
An exact proof term for the current goal is SNo_0.
Assume H1: 0 < x.
rewrite the current goal using recip_SNo_poscase x H1 (from left to right).
An exact proof term for the current goal is SNo_recip_SNo_pos x Hx H1.
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