 Brouwer actually posited the fan theorem without the restriction of detachability of the bar. Attempts to prove that more general fan theorem constructively rely on an analysis of how we could know that a subset is a bar, and led Brouwer to a notion of bar induction; this is discussed in Section 3.6 of the entry on  intuitionism in the philosophy of mathematics;  another good reference for bar induction is van Atten (2004). We shall return to fan theorems in Section 4. 