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. Proof, S. Let, ?. L. , S. ?. , ?. ?. Next et al., By construction of X(S), there exists a path w in X(S) By definition of the function X, we have that this path will be represented by the word w ? L ? . Now, let us consider two words u and v in L such that u ? v. By construction of X(S), u and v will be two paths of X(S) leading to the same vertex. By definition of the function X, the two words u and v in L ? will be equivalent regarding to the relation, ? viewed as subsets of (L ? L ? ) 2 . [S ? S(X(S))]: Let us consider w ? L)) ? S]: Let w ? ? L ? . By definition there exists a path ? ? in X(S) labeled by w ? from the pointed vertex to a vertex u. By Definition 59 there exists a word in L describing the path ? ? , hence w ? ? L. Similarly we prove the inclusion ? L ? ?? L