One of these things exists if the alternating sum of matrix traces induced by a map is non-zero. One of these things given by Lawvere’s (“law-VEER’s”) theorem is used in generalizations (10[1])of diagonal arguments. The existence of these things on certain triangulable spaces is guaranteed by a non-zero value of Lefschetz’s number. Functions on complete metric spaces (10[3])with Lipschitz constants (10[1])strictly less than one all have one of these (10[1])things, (10[2])a fact usually used (10[1])to prove (10[1])Picard’s (-5[1])theorem for the (10[1])well-posedness of ODEs (“O-D-E’s”). A theorem named for Stefan Banach gives conditions (10[1])for contractions (10[1])to have one of these things. (10[3])For a one-variable function “y equals g-of-x,” these things are found at the intersection (10[1])of the function g with the line y (10[1])equals x. (-5[1])For 10 points, a theorem named after Brouwer (“BRAO-wer”) gives the existence (10[1])of what kind of point satisfying f-of-x equals x? ■END■ (10[1]0[1])