 Note that this is equivalent to saying that "w" belongs to the normal closure of "R" in "F"("X"), that is, there exists a representation of "w" as

where "m" ≥ 0 and where "ri" ∈ "R"± 1 for "i" = 1, ..., "m".For "w" ∈ "F"("X") satisfying "w" = 1 in "G", the "area" of "w" with respect to (∗), denoted Area("w"), is the smallest "m" ≥ 0 such that there exists a representation (♠) for "w" as the product in "F"("X") of "m" conjugates of elements of "R"± 1.A freely reduced word "w" ∈ "F"("X") satisfies "w" = 1 in "G" if and only if the loop labeled by "w" in the presentation complex for "G" corresponding to (∗) is null-homotopic.