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Proofgold Proof

pf
Let x0 of type ι be given.
Assume H0: x0SNoS_ omega.
Apply SNoS_E2 with omega, x0, minus_SNo x0SNoS_ omega leaving 3 subgoals.
The subproof is completed by applying omega_ordinal.
The subproof is completed by applying H0.
Assume H1: SNoLev x0omega.
Assume H2: ordinal (SNoLev x0).
Assume H3: SNo x0.
Assume H4: SNo_ (SNoLev x0) x0.
Apply SNoS_I with omega, minus_SNo x0, SNoLev (minus_SNo x0) leaving 3 subgoals.
The subproof is completed by applying omega_ordinal.
Apply minus_SNo_Lev with x0, λ x1 x2 . x2omega leaving 2 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H1.
Apply SNoLev_ with minus_SNo x0.
Apply SNo_minus_SNo with x0.
The subproof is completed by applying H3.