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 $s\in S$, $t\in T$, $s\neq t$, $w<sw<stw\implies tw<stw$.

Where $<$ is the Bruhat ordering. Note that $sw<stw\implies w<tw$.

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).