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This was Nagal's first event on the ATP Tour since his first-round exit from the US Open.
Reserve player Aryan Shah has been drafted into the main team after Nagal's withdrawal, while Manas Dhamne has been put on standby.
Frenchman Corentin Moutet knocked out the Indian with a three-set win.
Nagal committed 44 unforced errors as he lost in four sets.
This will be Nagal's second appearance at the Games after he previously became the first Indian man since Leander Paes to win a singles match at the Tokyo Olympics.
Nagal lost to Russian Karen Khachanov 2-6 0-6 6-7(5).
Nagal threatened to spoil Rune's party by taking the second set after the second round match, which was suspended due to rain on Wednesday, resumed. However, he lost his serve twice in the final set.
Nagal became the first Indian player to win a singles main draw match at an ATP Masters event on clay.
Nagal is the first Indian to feature in singles top 100 since Prajnesh Gunneswaran made the cut in 2019.
Nagal began in the same dominating fashion but the 18-year-old wild card entrant got better as the game progressed to walk out a winner.
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