Let x0 of type ι be given.
Apply real_SNo with
x0.
The subproof is completed by applying H0.
Apply SNoLt_trichotomy_or_impred with
x0,
0,
recip_SNo x0 ∈ real leaving 5 subgoals.
The subproof is completed by applying L1.
The subproof is completed by applying SNo_0.
Apply recip_SNo_negcase with
x0,
λ x1 x2 . x2 ∈ real 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 x2 ∈ real.
Apply recip_SNo_0 with
λ x1 x2 . x2 ∈ real.
The subproof is completed by applying real_0.
Apply recip_SNo_poscase with
x0,
λ x1 x2 . x2 ∈ real 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.