I'm pretty sure this question has been asked before, I tried looking for something like it, however, I couldn't find anything.
Given two relations $R(A,B),S(B,C)$, assuming S is not empty, prove the following equality: $$(R \times \pi_A(S)) \div \pi_A(S) \equiv R$$
Intuitively:
It's pretty clear that we are concatenating each entry in $R$ with every entry in $\pi_A(S)$, thus when dividing we get the relation $R$.
Is my claim correct?, if so how can I prove this formally?