In a way that has been demonstrated or shown; provably or evidently.
From demonstrated (past participle of demonstrate) + -ly (adverb-forming suffix). This emphasizes that something has already been shown or proven.
Adding -ly to 'demonstrated' creates an adverb meaning 'in a demonstrated way,' but it's so awkward-sounding that almost nobody uses it—'demonstrably' does the same job with fewer syllables, showing how English favors economy of speech.
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