Uniqueness of tetration. Small lemma.

[Kouznetsov]

I try to simplify the proof of uniqueness of holomorphic tetration by bo198214.

Let \( u \in \mathbb{R} \) and \( v \in \mathbb{R} \) and \( v>0 \)
Let \( \mathcal{S}=\{z\in \mathbb{C} : u<\Re(z)<u+1+v} \)
Let \( \alpha \) be entire 1-periodic function.
Let \( \mathcal{D}=\alpha(\mathcal{S}) \).

According to the Picard’s Little Theorem [1],
either \( \alpha \) is trivial function (constant),
or \( \mathcal{D}=\mathbb{C} \) or \( \exists c \in \mathbb{C} \) such that \( \mathbb{C}=\mathcal{D} \cup c \).

Let \( \mu(z)=z-\alpha(z) \forall z \in \mathbb{C} \).
Let \( \mathcal{E}=\mu(\mathcal{S}) \)

Assume, \( \alpha \) is not constant.
How to prove that at least one of these two statements is true: \( -2\in \mathcal{E} \), \( -3\in \mathcal{E} \)
?

[1] Weisstein, Eric W. ”Picard’s Little Theorem.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/PicardsLittleTheorem.html