Proof of correctness of a homomorphic ElGamal sum
The question is not very clear about exactly what you want to prove and what is publicly known, but here's my answer, based on my best guess at what you mean:


Each party should publish $(R_1,S_1)$ and $(R_2,S_2)$.  They should also publish $(R_3,S_3)$.
Now anyone can verify that $(R_3,S_3)$ is a correctly-formed encryption of the sum of the messages corresponding to $(R_1,S_1)$ and $(R_2,S_2)$.  In fact, there is no need for any fancy zero-knowledge proof: given $(R_1,S_1)$ and $(R_2,S_2)$, anyone who is interested can re-compute $(R_3,S_3)$ on their own and verify that the result they obtained matches what was published.