A priest lives at the bottom of a middling hill. Once a year, he makes a pilgrimage by foot to a temple at the top of the hill. He makes it a point to set out at daybreak, walks up leisurely, stops for snacks and lunch and an occasional rest ... and reaches the temple just as the sun is setting.
He spends a few days at the temple, in prayer and meditation. Then he makes his return trip in much the same way. He sets out at daybreak, walks down leisurely, stops for snacks and lunch and so forth ... and reaches his home a little while before sunset, because walking downhill is faster than walking uphill.
Question: is there a point on the hill at which the priest can be found at the same time on both days? Yes, no or maybe? Whatever your answer, prove it.
(Note: If your answer is yes, I'm not interested in where the point is, or at what time he is there. Just prove that there is such a point).