Let $F$ be a free group and $w\in F$ a cyclically reduced word. Let $v$ be a non-trivial proper subword of $w$. Is it true that $v\notin \langle w^F\rangle$?



Is is true that a proper subword cannot lie in the normal closure of a word?
This is true. See Theorem 2 of [ON RELATORS AND DIAGRAMS FOR GROUPS WITH ONE DEFINING RELATION
BY
C. M. WEINBAUM](https://projecteuclid.org/journalArticle/Download?urlId=10.1215%2Fijm%2F1256052287&referringURL=https%3A%2F%2Fwww.google.com%2F&isResultClick=False).


