Let x0 of type ι be given.
Assume H1: x0 = 0 ⟶ ∀ x1 : ο . x1.
Apply SNoLt_trichotomy_or_impred with
x0,
0,
mul_SNo x0 (recip_SNo x0) = 1 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 . mul_SNo x0 x2 = 1 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H2.
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 H2.
Apply mul_SNo_minus_distrR with
x0,
recip_SNo_pos (minus_SNo x0),
λ x1 x2 . x2 = 1 leaving 3 subgoals.
The subproof is completed by applying H0.
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 L3.
Apply mul_SNo_minus_distrL with
x0,
recip_SNo_pos (minus_SNo x0),
λ x1 x2 . x1 = 1 leaving 3 subgoals.
The subproof is completed by applying H0.
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 L3.
Apply recip_SNo_pos_invR 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 L3.
Assume H2: x0 = 0.
Apply FalseE with
mul_SNo x0 (recip_SNo x0) = 1.
Apply H1.
The subproof is completed by applying H2.
Apply recip_SNo_poscase with
x0,
λ x1 x2 . mul_SNo x0 x2 = 1 leaving 2 subgoals.
The subproof is completed by applying H2.
Apply recip_SNo_pos_invR with
x0 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H2.