Let x0 of type ι be given.
Apply SNo_eps_ with
x0.
Apply nat_p_omega with
x0.
The subproof is completed by applying H0.
Apply SNo_eps_pos with
x0.
Apply nat_p_omega with
x0.
The subproof is completed by applying H0.
Apply mul_SNo_nonzero_cancel with
eps_ x0,
recip_SNo_pos (eps_ x0),
exp_SNo_nat 2 x0 leaving 5 subgoals.
The subproof is completed by applying L1.
Apply SNoLt_irref with
0.
Apply H3 with
λ x1 x2 . SNoLt 0 x1.
The subproof is completed by applying L2.
Apply SNo_recip_SNo_pos with
eps_ x0 leaving 2 subgoals.
The subproof is completed by applying L1.
The subproof is completed by applying L2.
Apply SNo_exp_SNo_nat with
2,
x0 leaving 2 subgoals.
The subproof is completed by applying SNo_2.
The subproof is completed by applying H0.
Apply mul_SNo_eps_power_2 with
x0,
λ x1 x2 . mul_SNo (eps_ x0) (recip_SNo_pos (eps_ x0)) = x2 leaving 2 subgoals.
The subproof is completed by applying H0.
Apply recip_SNo_pos_invR with
eps_ x0 leaving 2 subgoals.
The subproof is completed by applying L1.
The subproof is completed by applying L2.