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At age 22, Puig became Puerto Rico's first Olympic gold medallist with a stunning performance.
The Canadian teen had earlier stunned third seed and defending champion Naomi Osaka.
The former world number one was going through a low phase in her career when an interactive session with a Keralite yoga expert revived her fortunes.
Top seed Ash Barty became the first Australian woman to reach the Wimbledon singles final for 41 years.
In the first all-Australian women's Grand Slam quarterfinal since the 1980 Wimbledon championships, the 25-year-old Barty produced a calm and composed display on Centre Court.
Kerber's poor record at the French Open got no help when the German was knocked out in the first round by Ukrainian qualifier Anhelina Kalinina.
Jelena Ostapenko became the first Latvian to reach the Wimbledon singles semifinals after she subdued Slovakia's Dominika Cibulkova 7-5, 6-4.
Angelique Kerber will face either seven-times champion Serena Williams or fellow German Julia Georges in the final.
Seven-time champion Serena had been bidding to claim a record-equalling 24th Grand Slam crown but made too many errors as Kerber ran out a comfortable winner.
Former coach of Kim Clijsters (2009-11), current world No. 1 Simona Halep (2014) and Victoria Azarenka (2015-16), Fissette joined forces with Kerber in November, 2017.
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