Let CS be a combined scheme of $n$ public key subschemes.
CS is composed of two algorithms Setup and KeyGen, that all the subschemes share, plus all the other algorithms of each subscheme.
Suppose that each one of the subschemes is individually secure in the Random Oracle Model (ROM).
I want to prove that CS is secure, with the following definition: CS is secure if all its subschemes are jointly secure, meaning that each subscheme is secure in the presence of the others.
$Proof$. Suppose an adversary A is able to break the security of subscheme $i \in n$ in the presence of the others.
We can construct an adversary B that simulates subscheme $j \neq i \in n$ by programming random oracles, in a game that is indistinguishable from the real experiment.
B will then use the attack of A to break the standalone security of subscheme $i$, which we assumed to be secure. We prove, by contradiction, that CS is secure.
Does this make sense? If not, how can I prove that CS is secure?