---
tag: XGZ08QFJV8
kind: statement
role: lemma
name: Strengthened version of Van Kampen's lemma
sources:
  - wikipedia: en:18871460@r1368818220
    locator: Van Kampen diagram, section Van Kampen lemma > Strengthened version of Van Kampen's lemma
    note: adopted
    mention: wikipedia://en/18871460#TKQG
    sections: [Statement]
inventory: [BQRJM26XKN]
---

## Statement

Moreover, the above proof shows that the conclusion of Van Kampen's lemma can be strengthened as follows. Part 1 can be strengthened to say that if $\mathcal D\,$ is a Van Kampen diagram of area *n* with boundary label *w* then there exists a representation (♠) for *w* as a product in *F*(*A*) of exactly *n* conjugates of elements of *R*<sub>∗</sub>. Part 2 can be strengthened to say that if *w* is freely reduced and admits a representation (♠) as a product in *F*(*A*) of *n* conjugates of elements of *R*<sub>∗</sub> then there exists a reduced Van Kampen diagram with boundary label *w* and of area *at most* *n*.