G. Let and ?. ?. , Then ?(tv) is the identity of G and thus tv ? Z. Since S is a tree set, it is acyclic and thus Z * is saturated in S by Theorem 5.12 Thus This implies that tv ? Z * . Since tv ? A * t, we have f (r)uf (s)v = f (r)qf (s) and thus uf (s)v = qf (s) for some q ? S. Since Z * is right unitary, f (r), f (r)uf (s)v ? Z * imply uf (s)v = qf (s) ? Z * . In turn, since Z * is left unitary, qf (s), f (s) ? Z *

. Proof, Since X is a prefix code, W is a prefix code. Since X is v ?1 S-maximal, W is ? ?1 (v ?1 S)-maximal by Proposition 2.9 (ii) and thus H-maximal since

?. Let and ?. W. , Then ?(xy), ?(y) ? X imply ?(x) ?

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