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The phrase "after the first set" is correct and usable in written English.
It can be used in contexts such as sports, games, or any sequential activities where a first set is followed by subsequent sets or actions.
Example: "The players took a short break after the first set before continuing with the match."
Alternatives: "following the initial set" or "subsequent to the first set".
Exact(56)
"He leaves just after the first set ends.
Most of Ahmet's guests left after the first set.
"After the first set I couldn't move anymore.
After the first set, I was like, 'No, please, no.
I gave him very few opportunities after the first set.
"After the first set, I really settled in and got comfortable".
"But I began to play much better after the first set".
A brilliant last two sets from Williams who looked dead and buried after the first set.
After the first set, two boats turned away, but the vendor's boat did not.
"I was kind of kicking myself a little bit after the first set," said Sampras.
"Nothing really worked after the first set," said Sharapova, who served 17 double faults.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com