$L(x)$ means "person $x$ from our town loves lolicon."
$G(x)$ means "person $x$ from our town loves guro."
Both of $L(x)$ and $G(x)$ share the same universe.
The equivalency in question is $$\exists x L(x) \lor \exists x G(x) \equiv \exists x ( L(x) \lor G(x) )$$
I translate the left part as "Somebody in our town loves lolicon and/or somebody in our town loves guro."
I translate the right part as "Somebody in our town loves lolicon and/or loves guro."
Let's assume that there are only two such freaks in our little town, one loves lolicon and the other one loves guro.
The left side of the equivalency would be true under such assumption. But I worry for the right side.
My reasoning is following:
It seems for me that we can interpret $\exists x ( L(x) \lor G(x) )$ as prediction of tastes for the whole group of freaks that has size $\ge 1$. "There is a group consisting of one or more person(s) in our town whose members all either love lolicon or love guro or love both guro and lolicon". So if this group consists of two persons, then either both of them love guro or both of them love lolicon or both of them love guro and lolicon at the same time. There can't be a situation when one loves guro and the other one loves lolicon.
I would like to see my if reasoning is true or false. And why.