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Hamilton's win at a sunny Spa-Francorchamps circuit on Sunday, which left him two short of Michael Schumacher's record tally of 91 victories, was his fifth in seven races.
Seven of the 10 F1 teams have joined forces with leading aerospace and engineering companies to ramp up production of ventilators while Mercedes has also worked with medics and academics to produce an alternative breathing aid.
The Mercedes driver triggered initial concern when he started a series of posts by saying he felt like 'giving up on everything.'
The result also left only Bottas capable of challenging Hamilton, who saw his lead narrow to 64 points, in the overall standings.
The victory was the 82nd of Hamilton's career and ninth of the season.
McLaren, who have used Renault engines since last year, have clinched a deal with Mercedes until at least the end of 2024, resurrecting a partnership that ran from 2005 to 2014.
A spokesman for the governing body FIA said the International Court of Appeal in Paris had found the Swiss-based team's protest was not admissible.
The season will end with the Abu Dhabi Grand Prix at the Yas Marina track on December 13. It is unclear whether Hamilton will take part.
French driver Romain Grosjean had a miraculous escape from a fiery first-lap crash that ripped his car in two.
The 35-year-old Mercedes driver matched Ferrari great Michael Schumacher's seven titles in style, lapping sole rival and teammate Valtteri Bottas on a nightmare afternoon for the spinning Finn.
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