We show that the two following axioms are equivalent:
TOR : For every linearly ordered set, there exists an ordinal o, and a strictly increasing mapping j from X to the lexicographic order {0,1}^o.
CE: For every complete linearly ordered set X, there is a mapping * which associates to every non-empty open convex subset ]a,b[ of X, an element a*b in ]a,b[, such that:
for every a, b, c in X, if b is in ]a,c[ then a*c=a*b or a*c=b*c or a*c=b.

This solves a question raised in my thesis (see [Mo88b]).