Van Kampen Diagram - Van Kampen Lemma

A key basic result in the theory is the so-called van Kampen lemma which states the following:

1. Let be a van Kampen diagram over the presentation (†) with boundary label w which is a word (not necessarily freely reduced) in the alphabet A ∪ A−1. Then w=1 in G.
2. Let w be a freely reduced word in the alphabet A ∪ A−1 such that w=1 in G. Then there exists a reduced van Kampen diagram over the presentation (†) whose boundary label is freely reduced and is equal to w.