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Let us assume, as usual, that $I=[r,s],\ J=[u,v],\ K=[p,q]$.


Lemma

(IJ)K is the smallest interval containing all (xy)z where $x\in I$, $y\in J$, $z\in K$.


\begin{proof} First note that $IJ$\space contains all elements of the form $xy$,... ...ce contains them by assumption, so $W$\space contains all of $(IJ)K$.\end{proof}

 
Mark Bridger
1/23/2002