A more concise proof of Sen's theorem can be based on the following lemma:

LEMMA. If xPyPz holds, then someone ranks x above the other two, someone ranks y between the other two, and someone ranks z below the other two.

(b) Use the lemma to prove Sen's theorem. (Hint: Assume that P is not transitive. Then there are alternatives a, b, and c so that aPbPcPa holds. Notice that any of the alternatives a, b, or c can play the role of any one of the alternatives x, y, z in the lemma.