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  "body": "\n## Statement\n\nMoreover, 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*\u003csub\u003e∗\u003c/sub\u003e. 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*\u003csub\u003e∗\u003c/sub\u003e then there exists a reduced Van Kampen diagram with boundary label *w* and of area *at most* *n*.\n",
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