section we study relations between derivated words of a Sturmian word and derivated words of its preimage under one of the morphisms $\varphi_a, \varphi_b, \varphi_\alpha$ and $\varphi_\beta$. We prove that the set of all derivated words of these two infinite words coincide up to at most one derivated word, see Theorems \[thm:preimages\_fi\_b\] and \[thm:preimages\_fi\_a\]. This will be crucial fact for proving the main results of this paper. Because of , the roles of $\varphi_a$ and $\varphi_\alpha$ and, analogously, the roles of $\varphi_b$ and $\varphi_\beta$ are symmetric. Therefore we can restrict the statements and proofs in this section to the morphisms $\varphi_a$ and $\varphi_b$ with no loss of generality. Again we use results from [@Lo83], in particular this slightly modified Proposition 2.3.2:

Let $\bf x$ be an infinite word.

(i) If $\varphi_b(\bf x)$ is Sturmian, then $\bf x$ is Sturmian.

(ii) If $\varphi_a(\bf x)$ is Sturmian and $\bf x$ starts with the letter $1$, then $\bf x$ is Sturmian.

\[thm:preimages\_fi\_b\] Let $\uu$ and $\uu'$ be Sturmian words such that $\uu = \varphi_{b}(\uu')$. Then the sets of their derivated words satisfy $$\Der(\uu) = \Der(\uu') \cup \{ \uu'\}\,.$$

The proof of the previous theorem is split into two parts: In Proposition \[prop:der\_of\_images\_fi\_b\], Item 