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When trying to prove an existential statement [math]\href{/cs2800/wiki/index.php/%E2%88%83}{∃x}, \href{/cs2800/wiki/index.php/Predicate}{P}(x) [/math], you need to give a specific value of [math]x [/math] (a witness).

Often, in a proof, it is not immediately obvious what the witness should be. Finding one often involves solving some equations or combining some known values.

One nice technique for finding a witness is to simply leave a blank space for the value of [math]x [/math] and continue on with your proof of [math]P(x) [/math]. As you go, you may need [math]x [/math] to satisfy certain properties (for example, maybe you need [math]x \gt 17 [/math] at one point, and later you need [math]x \lt 85 [/math]). You can make a "wishlist" on the side of your proof, reminding you of all the properties you want [math]x [/math] to satisfy. Once you've completed your proof, you can go back and find a specific value of [math]x [/math] (say, [math]50 [/math]) that satisfies all of your wishes.