Equivalence relations 
Having trouble proving this is an equivalence relation.
Is it suffice to say that let $x y z$ be any string in $\Sigma^*$, 
$(xz \in L \iff yz \in L) \rightarrow (yz \in L \iff xz \in L)$ shows that $xRy \rightarrow yRx$?

The specific situation of formal languages is unnecessary.
Suppose there is a relation called "?".  Given $x$, define $??(x)$ be the set of $u$ such that $x?u$.  Let $???$ be the relation "has the same value of ?? as", so that $a???b$ when $??(a)=??(b)$.  Then ??? is clearly an equivalence relation.  That's all that is going on here, for a particular choice of $?$.
