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Osaka, who beat Belarusian Alejandra Sassanovic 6-2, 6-3, reached the quarterfinals for the first time as winner of the qualifying match of Ukraine, Katerina Kozlova and Switzerland's Belinda Pencic.
Halib, who is preparing to defend his title in French at Roland Garros, defeated Slovakia's second seed Viktoria Kuzmova 6-0, 6-0 in 44 minutes.
Halib, who is looking for a third title in Madrid after being crowned twice in 2016 and 2017, will face a difficult task in the quarterfinals: the Australian Ashley Party (ninth in the world) and the world's number one, Yulia Botintseva (43) will face three sets 4-6, 6-1, 6 2
Seventh-seeded Kiki Pertzen of the Netherlands beat Latvia's Anastasia Sevastova 13-1 6-1 6-2. The next match could be a comeback in the final last year if Czech Petra Kvetova beat France's Caroline Garcia.
Men
In the second round of the Masters Masters (1000 points), the sixth-seeded Japanese, Kih Nishikori, needed more than two hours to leave the Hungarian qualifier Hugo Dalian 7-5 to take on Switzerland's Stanislaw Wawrinka. 3, 6-4.
Frenchman Jeremy Scharde of France beat third seed Diego Schwartzman 6-1, 6-2 to meet Novak Djokovic of Serbia.
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