# Consequence of Lifting property of Bruhat ordering

I am reading the book: Anders Björner, Francesco Brenti --- Combinatorics of Coxeter Groups.

I would like to know whether a variation of Corollary 2.2.8 is true. In other words, does the following holds:

For $$s\in S$$, $$t\in T$$, $$s\neq t$$, $$w.

Where $$<$$ is the Bruhat ordering. Note that $$sw.

The assertion is true under the assumption $$\ell(tw)=\ell(w)+1$$ by the proof of Corollary 2.2.8.

A proof of Corollary 2.2.8 can be found on James E. Humphreys---Reflection Groups and Coxeter Groups (Lemma 5.11).

No, it is not true. Take $$\mathfrak{sl}_3$$ with simple reflections $$r, s$$. Then the longest element $$t := rsr=srs$$ is also a reflection. Put $$w=e$$.
Then $$w=e < sw=s < stw = rs$$, but $$tw = srs \nless stw = rs$$.