In this theorem a real number can play one of two roles. Sometimes a real number plays the usual role of a real number and other times a real number plays the role of a vector in the vector space R.
This kind of ambiguity is not unusual -- for example, you probably play several roles. When you go into a store you play the role of a customer. In one classroom you might play the role of a student and in another one you might play the role of a teacher. Sometimes you play two roles in the same scene -- for example you might play the role of a cook and also play the role of a diner when you cook dinner for yourself and a few friends.