Showing an equivalent definition of $CO-RE$
Equivalently: $\overline{L_1}$ is RE iff there is some $\overline{L_2}$ that is $R$ so that $\overline{L_1}=\{u  | \exists v : &lt;u,v&gt;\in \overline{L_2}\}$ is RE.


$\Rightarrow$


Take $\overline{L_2}=\{&lt;u,v&gt; | v\text{ represents an execution accepting u in the MT accepting }\overline{L_1}\}$. It's R because you just simulate the MT following v and it'll either work or not. And it vefiries the property $\overline{L_1}=\{u  | \exists v : &lt;u,v&gt;\in \overline{L_2}\}$ because a word is accepted iff some execution accepts.

You could also take $\overline{L_2}=\{&lt;u,n&gt; | u\text{ is accepted by an execution of size} n\}$.


$\Leftarrow$


Enumerate all $v$, for each, test is $&lt;u,v&gt;\in\overline{L_2}$ and if it is, accept. If the word is in $\overline{L_1}$ then some $v$ will work so it'll be accepted. So $\overline{L_1}$ is indeed RE.



Your property says is that being RE is the same thing as being the projection of a R set.