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  "body": "\n## Statement\n\nA key basic result in the theory is the so-called *Van Kampen lemma* which states the following:\n\n1. Let $\\mathcal D\\,$ 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*\u003csup\u003e−1\u003c/sup\u003e. Then *w*=1 in *G*.\n1. Let *w* be a freely reduced word in the alphabet *A* ∪ *A*\u003csup\u003e−1\u003c/sup\u003e such that *w*=1 in *G*. Then there exists a reduced Van Kampen diagram $\\mathcal D\\,$ over the presentation (†) whose boundary label is freely reduced and is equal to *w*.\n\nFirst observe that for an element *w* ∈ *F*(*A*) we have *w* = 1 in *G* if and only if *w* belongs to the normal closure of *R* in *F*(*A*) that is, if and only if w can be represented as\n\n$w=u_1s_1u_1^{-1}\\cdots u_n s_nu_{n}^{-1} \\text{ in } F(A),$ (♠)\n\nwhere *n* ≥ 0 and where *s\u003csub\u003ei\u003c/sub\u003e* ∈ *R*\u003csub\u003e∗\u003c/sub\u003e for *i* = 1, ..., *n*.\n\nPart 1 of Van Kampen's lemma is proved by induction on the area of $\\mathcal D\\,$. The inductive step consists in \"peeling\" off one of the boundary regions of $\\mathcal D\\,$ to get a Van Kampen diagram $\\mathcal D'\\,$ with boundary cycle *w* and observing that in *F*(*A*) we have\n\n$$w=usu^{-1} w',\\,$$\n\nwhere *s*∈*R*\u003csub\u003e∗\u003c/sub\u003e is the boundary cycle of the region that was removed to get $\\mathcal D'\\,$ from $\\mathcal D\\,$.\n\nThe proof of part two of Van Kampen's lemma is more involved. First, it is easy to see that if *w* is freely reduced and *w* = 1 in *G* there exists some Van Kampen diagram $\\mathcal D_0\\,$ with boundary label *w*\u003csub\u003e0\u003c/sub\u003e such that *w* = *w*\u003csub\u003e0\u003c/sub\u003e in *F*(*A*) (after possibly freely reducing *w*\u003csub\u003e0\u003c/sub\u003e). Namely consider a representation of *w* of the form (♠) above. Then make $\\mathcal D_0\\,$ to be a wedge of *n* \"lollipops\" with \"stems\" labeled by *u\u003csub\u003ei\u003c/sub\u003e* and with the \"candys\" (2-cells) labelled by *s\u003csub\u003ei\u003c/sub\u003e*. Then the boundary label of $\\mathcal D_0\\,$ is a word *w*\u003csub\u003e0\u003c/sub\u003e such that *w* = *w*\u003csub\u003e0\u003c/sub\u003e in *F*(*A*). However, it is possible that the word *w*\u003csub\u003e0\u003c/sub\u003e is not freely reduced. One then starts performing \"folding\" moves to get a sequence of Van Kampen diagrams $\\mathcal D_0, \\mathcal D_1, \\mathcal D_2,\\dots\\,$ by making their boundary labels more and more freely reduced and making sure that at each step the boundary label of each diagram in the sequence is equal to *w* in *F*(*A*). The sequence terminates in a finite number of steps with a Van Kampen diagram $\\mathcal D_k\\,$ whose boundary label is freely reduced and thus equal to *w* as a word. The diagram $\\mathcal D_k\\,$ may not be reduced. If that happens, we can remove the reduction pairs from this diagram by a simple surgery operation without affecting the boundary label. Eventually this produces a reduced Van Kampen diagram $\\mathcal D\\,$ whose boundary cycle is freely reduced and equal to *w*.\n",
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