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1. To show the validity of something. A noun or pronoun can be used between "prove" and "out." If you can prove out these allegations, that company will soon be faced with a scandal.
2. To succeed. A noun or pronoun can be used between "prove" and "out." My hypothesis still has not proved out, and I can't determine why.
Succeed, turn out well, as in Farm-raised trout has proved out so well that the fish industry plans to experiment with other species . [Mid-1900s]