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MathLearner123

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I want to prove following (Big Picard Theorem forms):\

The followings are equivalent:\

I have proved that

**Theorem**.The followings are equivalent:\

**a)**If ##f \in H(\mathbb{D}\setminus\{0\})## and ##f(\mathbb{D}') \subset \mathbb{C} \setminus \{0, 1\}##, then ##f## has a pole of an removable singularity at ##0##.\**b)**Let ##\Omega \subset \mathbb{C}## is a open subset, ##f : \Omega \to \mathbb{C}## is holomorphic and ##z_0 \in \mathbb{C}##. If ##f## has an essential singularity at ##z_0##, then, with at most one exception, ##f## attains every complex value infinitely many times;\**c)**Let ##f : \mathbb{C} \to \mathbb{C}## a entire function which is not polynomial. Then, with at most one exception, ##f## attains every complex value infinitely many times;I have proved that

**a)**##\implies##**b)**##\implies##**c)**and that**b)**##\implies##**a)**but I don't know how to start proving that**c)**implies**a)**or**b)**. And another thing: Is mathematically correct to say that those points are equivalent? Thanks!!
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