Englishfor English speakers
existence
Noun
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The existence of a thing is the quality of being real, of being in the world, of just being.
Do you believe in the existence of God?
The existence of a mark on my shirt is the reason I didn't go to the party.
proof
Noun
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Proof is evidence which shows something is correct.
He has proof that he was not at the crime scene. A witness says that they were somewhere else.
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A proof can show how a theorem is correct.