Let x0 of type ι be given.
Apply SNoLt_trichotomy_or_impred with
x0,
0,
recip_SNo (recip_SNo x0) = x0 leaving 5 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying SNo_0.
Apply recip_SNo_negcase with
x0,
λ x1 x2 . recip_SNo x2 = x0 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
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 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 recip_SNo_negcase with
minus_SNo (recip_SNo_pos (minus_SNo x0)),
λ x1 x2 . x2 = x0 leaving 3 subgoals.
Apply minus_SNo_Lt_contra1 with
0,
recip_SNo_pos (minus_SNo x0) leaving 3 subgoals.
The subproof is completed by applying SNo_0.
The subproof is completed by applying L3.
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.
Apply minus_SNo_invol with
recip_SNo_pos (minus_SNo x0),
λ x1 x2 . minus_SNo (recip_SNo_pos x2) = x0 leaving 2 subgoals.
The subproof is completed by applying L3.
Apply recip_SNo_pos_invol with
minus_SNo x0,
λ x1 x2 . minus_SNo x2 = x0 leaving 3 subgoals.
Apply SNo_minus_SNo with
x0.
The subproof is completed by applying H0.
The subproof is completed by applying L2.
Apply minus_SNo_invol with
x0.
The subproof is completed by applying H0.
Assume H1: x0 = 0.
Apply H1 with
λ x1 x2 . recip_SNo (recip_SNo x2) = x2.
Apply recip_SNo_0 with
λ x1 x2 . recip_SNo x2 = 0.
The subproof is completed by applying recip_SNo_0.
Apply recip_SNo_poscase with
x0,
λ x1 x2 . recip_SNo x2 = x0 leaving 2 subgoals.
The subproof is completed by applying H1.
Apply recip_SNo_poscase with
recip_SNo_pos x0,
λ x1 x2 . x2 = x0 leaving 2 subgoals.
Apply recip_SNo_pos_pos with
x0 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply recip_SNo_pos_invol with
x0 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.