Let x0 of type ι be given.
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 recip_SNo_negcase with
x0,
λ x1 x2 . SNoLt x2 0 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply minus_SNo_Lt_contra1 with
0,
recip_SNo_pos (minus_SNo x0) leaving 3 subgoals.
The subproof is completed by applying SNo_0.
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 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.