Dist. Ct. did not err in denying plaintiff’s motion in instant Anti-Injunction Act claim to enjoin state court from proceeding in defendant’s underlying lawsuit against plaintiff to enforce promissory note and guaranties, even though: (1) defendant had previously filed same claim on note and guaranties in federal court, which had granted plaintiff’s motion to dismiss said claim without prejudice to filing amended federal complaint; (2) instead of filing amended complaint in federal court, defendant filed notice of voluntary dismissal under Rule 41(a)(1)(A)(i), which was granted by federal court; and (3) state court denied plaintiff’s motion to dismiss said lawsuit even though instant plaintiff had argued that prior federal court’s dismissal with leave to amended complaint precluded state court action on grounds of res judicata. Instant request for injunction was improper under Full Faith and Credit Act, where plaintiff had raised and lost its res judicata argument in state court. Moreover, party seeking injunction under exception to Anti-Injunction Act must show that state court preclusion was clear “beyond peradventure,” and plaintiff could not make such showing where, under Illinois law, federal court’s dismissal without prejudice was not final decision that was subject to res judicata defenses. Ct. rejected plaintiff’s contention that prior voluntary dismissal should have been considered as precluding subsequent state court action, even though plaintiff argued that: (1) instant Rule 41(a) motion to voluntarily dismiss was filed only after federal court had rendered “adverse ruling” that dismissed complaint without prejudice; and (2) under doctrine of springing finality, federal court’s initial dismissal without prejudice to filing amended complaint ripened into final dismissal on merits where defendant had ignored prior federal court’s instruction to file amended complaint.