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Proofgold Proof

pf
Let x0 of type ι be given.
Assume H0: x0SNoS_ omega.
Assume H1: ∀ x1 . x1x0∃ x2 . and (x2omega) (∀ x3 . x3SNoS_ omegaSNoLt (add_SNo x1 (minus_SNo (eps_ x2))) x3SNoLt x3 (add_SNo x1 (eps_ x2))x3x0).
Apply andI with x0SNoS_ omega, ∀ x1 . x1x0∃ x2 . and (x2omega) (∀ x3 . x3SNoS_ omegaSNoLt (add_SNo x1 (minus_SNo (eps_ x2))) x3SNoLt x3 (add_SNo x1 (eps_ x2))x3x0) leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.