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Cox got off to a bad start on Court 18 when he lost his opening service game, soon falling 5-2 behind before Chardy won the set 6-2 in just 23 minutes.
It looked like Chardy would break at the start of the second set, but Cox saved four break points and remained strong on serve for the remainder of the set.
A tie-break was needed, with Chardy coming through 7-3 as Cox's serve faltered at a crucial stage.
The duo then traded breaks at the start of the third before another breaker was required. This time it was Cox who got himself on the board by taking his third set point at 7-6 to the delight of the crowd.
Lincoln-born Cox couldn't complete a turnaround, though, as he was broken four times in the fourth set by Chardy, who took it 6-3 to advance.