Strengthened version of Van Kampen's lemma

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_∗. 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_∗ then there exists a reduced Van Kampen diagram with boundary label w and of area at most n.