i'm in trouble trying to prove the following proposition:
Proposition If $\ C$ is a class closed under reductions and a $\ C$-Complete problem belongs to $\ coC$, then $C=coC$.
Note: $C$ can be any class.
To prove that proposition i thought to split the proof in two ways:
(*): $C \subseteq coC$
I suppose it exists such an $L \in coC$ that is $C$-Complete. For all $L' \in C$ there is a reduction $R : L \rightarrow L'$. Thus $L \in coC$ and $coC$ is closed under reductions (because $C$ is), $L' \in coC$ also, so $C \subseteq coC$.
(*): $coC \subseteq C$
That's the part i'm not able to prove.
Thank you so much.