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Proofgold Proof

pf
Let x0 of type ι be given.
Assume H0: x0 ∈ SNoS_ omega.
Apply SNoS_E2 with omega, x0, finite (SNoR x0) leaving 3 subgoals.
The subproof is completed by applying omega_ordinal.
The subproof is completed by applying H0.
Assume H1: SNoLev x0 ∈ omega.
Assume H2: ordinal (SNoLev x0).
Assume H3: SNo x0.
Assume H4: SNo_ (SNoLev x0) x0.
Apply Subq_finite with SNoS_ (SNoLev x0), SNoR x0 leaving 2 subgoals.
Apply SNoS_finite with SNoLev x0.
The subproof is completed by applying H1.
Let x1 of type ι be given.
Assume H5: x1 ∈ SNoR x0.
Apply SNoR_E with x0, x1, x1 ∈ SNoS_ (SNoLev x0) leaving 3 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H5.
Assume H6: SNo x1.
Assume H7: SNoLev x1 ∈ SNoLev x0.
Assume H8: SNoLt x0 x1.
Apply SNoS_I2 with x1, x0 leaving 3 subgoals.
The subproof is completed by applying H6.
The subproof is completed by applying H3.
The subproof is completed by applying H7.
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