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Translation of "증명했어요" in English

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Examples of using "증명했어요" in a sentence and their english translations:

그런데 그들이 틀렸다는 걸 증명했어요.

Well, I guess we've proven them wrong.

우리는 우리의 이야기들이 가치가 있음을 증명했어요.

We proved our stories have value.