Let x0 of type ι be given.
Apply SNoS_E2 with
omega,
x0,
x0 = 0 leaving 3 subgoals.
The subproof is completed by applying omega_ordinal.
The subproof is completed by applying H0.
Apply real_SNoS_omega_prop with
0,
x0 leaving 3 subgoals.
The subproof is completed by applying real_0.
The subproof is completed by applying H0.
Let x1 of type ι be given.
Apply minus_SNo_0 with
λ x2 x3 . SNoLt (abs_SNo (add_SNo x0 x3)) (eps_ x1).
Apply add_SNo_0R with
x0,
λ x2 x3 . SNoLt (abs_SNo x3) (eps_ x1) leaving 2 subgoals.
The subproof is completed by applying H4.
Apply H1 with
x1.
The subproof is completed by applying H6.