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Reims, who once again played nicely but lacked sharpness in front of goal, are 14th on seven points after reducing the arrears through Alexis Flips.
Yilmaz delivered a perfect cross for David at the far post and the Canadian put the hosts in front with a header.
Lille were then on the back foot but made the most of their chances and Andre made it 2-0 two minutes before the break as he headed home from Yilmaz's corner.