In ZFC, every linear ordered space respect to the order topology is completely normal. I saw the this proof and proof of this statement in the book "Counterexamples of topology" (Example 39). But as I seen every proof of this statement uses choice. Even if (as I know) the proof of "Every linear continuum is normal" uses axiom of choice.
So I think that choice is essential to prove this statement. That is true? Thanks for any help.