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The women's doubles pair of Ashwini Ponnappa and N Sikki Reddy also entered the pre-quarterfinals with a hard-fought win.
Satwiksairaj Rankireddy and Chirag Shetty beat Chinese Taipei's Lee Jhe-Huei and Yang Po-Hsuan 27-25, 21-17 in 43 minutes.
The 12th-seed beat Pablo Abianof Spain 21-12, 21-16 in 36 minutes in the first round.
The US-headquartered tour's decision to walk away from one of its biggest markets was applauded by many leading figures in the tennis world but could cost the WTA hundreds of millions of dollars in broadcasting and sponsorship revenue.
The tennis legend has been a vocal critic of Indian Prime Minister Narendra Modi.
"What they whiffed through, what they shared...their stories and their success, that's something that lasts forever," said Hingis, who has partnered both Paes and Bhupathi.
US Open runner-up Leylah Fernandez jumped 45 spots, climbing from No.73 to No.28.
All the cheers and standing ovations in the world could not save Federer from the Centre Court mauling as it seemed his 39-year-old body finally said enough is enough.
The Czech downed little-known Swiss Viktorija Golubic in straight sets.
Iga Swiatek, who became the first Polish player to win a Grand Slam singles title when she beat Kenin in the French Open final, was named the WTA's Most Improved Player of the Year.