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The Wimbledon champion clinched her second straight W&S semifinal berth with a forehand deep into the corner that Kuznetsova could only steer into the net.
Muguruza needed 2 hours, 45 minutes - the longest of her three tournament matches - to clinch the win. It came less than 24 hours after she spent 2 hours, 18 minutes on the court and fought off three match points to overcome Madison Keys in the round of 16.
Kutznetsova, a two-time Grand Slam champion and the draw's oldest woman at 32, spent eight more minutes on the court in this match than she did in her previous two combined matches.
Top-ranked Karolina Pliskova topped Camila Giorgi 6-3, 4-6, 6-0.
In men's action, Nick Kyrgios shook off losing the first set to reach his first W&S quarterfinal with a 4-6, 7-6 (6), 6-3 win over Ivo Karlovic in the completion of a round-of-16 match postponed by rain from Thursday.
Kyrgios improved to 4-0 in his career against Karlovic. He'll face the winner of the postponed match between top-seeded Rafael Nadal and Albert Ramos-Vinolas.
Wait times can vary significantly based on the applicant's passport and the country from which they are applying
Lebanese Prime Minister Najib Mikati appealed to the international community to put pressure on Israel for a ceasefire
Generally, commission accounts for 2 per cent of the sale value and 5 per cent of the annual rental, excluding 5 per cent value-added tax
The Roads and Transport Authority offers multiple ways for motorists to pay their parking fees without having to do it physically
Year of war generates at least 42 million tonnes of rubble; UN estimate shows debris clearance might cost $1.2 billion; Piled up, rubble would fill Great Pyramid of Giza 11 times
The leader also approved the establishment of the National Agricultural Centre
RTA revealed that a total of 15.29 million transactions were made via smart apps alone
Israel is intensifying its attacks both in Gaza and Lebanon as the fight approaches its one-year mark