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Demonstrably

/dɪˈmɑnstrəbli/
B2
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(adv) In such a manner that is capable of being demonstrated, shown, or proved (without doubt).

Old beliefs die hard even when demonstrably false.
If you want to make anything demonstrably worse, add so-called "AI."
CEFR Practice Quiz
Old beliefs die hard even when ____ false.

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