Pe{x),y,s• Since, for every .y, WojS(x)W\yS( <iex 7, there is no .y

If Wa is coinfinite, then Ba is maximal. Take any k such that Ba C W^. Since k < | Wa | there is a string co^i . •. and a set U C Ba such that Ba UU is cofinite and Wk(y) = Ck on all y G U. It follows that Wk is a finite variant of Ba if Ck = 0 and Wk is cofinite if Ck = 1. So Ba is maximal.