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Tsonga won the first set 6-1, but was soon behind as Melzer took the second and third 6-3.
Tsonga started the fourth with a break and wrapped it up 6-2 before their match was postponed at 5-4 on No. 1 Court, with Tsonga up a break and ready to serve out the match.
He duly did so today, sealing the victory with an ace to reach the second round, where is likely to face big-serving American Sam Querrey.