Question: The task is to depict the following set:

$$\left\{ <x,y> \in \mathbb{R^2} \, |\, x^2 + y^2 > 1 \Rightarrow \exists z \, (x^2 + (y − z)^2 \leq \frac{1}{4}) \right\}$$

Answer: 
The second equality is not true in $\mathbb{C}$, but only in $\mathbb{R}$. See [this](https://en.wikipedia.org/wiki/Exponentiation#Failure_of_power_and_logarithm_identities).
This is analogous to: 

$$\cos 0 = \cos{2\pi}\implies \pi = 0$$