and requires substantial analysis, making use of some of the technical results from [@C].

**Initial Observations**

1.  For small values of $|r|$, the result is elementary by inspection.

2.  The general case will follow, via the normal form theorem for free products, from the case when $S \cup T = \hbox{Supp}(r)$ and so we can always assume the latter.

3.  When $|\hbox{Supp}(r)| = 2$, the conclusion is immediate, so that we can assume that $|\hbox{Supp}(r)| \ge 3$.

4.  If $S \subset T$ then $gF(S)g^{-1}\cap F(T) \subseteq gF(T)g^{-1}\cap F(T)$ which, by [@Ba], is cyclic unless $g \in F(T)$ so that we can always assume that $S\cap T$ is a proper subset of both $S$ and $T$;

5.  (assuming (iv)) If we write $B= S\cap T$ and then choose $A$ and $C$ disjoint so that $S = A\cup B$ and $T=B\cup C$, then the general case reduces to the case when $A$ and $C$ are singletons, say $A = \{a\}$ and $C=\{c\}$.

We therefore take all of these as given and embark upon the inductive case; our strategy will always be to assume the conclusion false and then work our way to a contradiction. In particular we shall assume that there exist 