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As soon as she finished, she lay down for a nap.
Literal
She [topic-は] finished [the moment-途端に] nap do [purpose-ために] lay-down.
~た途端に (the moment X happened) captures the immediacy — the instant she finished, she lay down. 仮眠する (to take a short nap) is a compound: 仮 (temporary) + 眠 (sleep). するために (in order to) marks the purpose. 横になる (to lie down) literally means 'to become horizontal' — a natural, common expression for lying down. The sentence paints a picture of exhaustion: the moment the task ended, she collapsed for rest.