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 and continue on with your proof of . As you go, you may need to satisfy certain properties (for example, maybe you need at one point, and later you need ). You can make a "wishlist" on the side of your proof, reminding you of all the properties you want to satisfy. Once you've completed your proof, you can go back and find a specific value of (say, ) that satisfies all of your wishes.