Relations: How to prove $R^2R^3 = R^5$?

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Relations: How to prove $R^2R^3 = R^5$ ?

I tried to go by this definition but I'm not quite sure I'm in the right path.

$RS = \{(x,y) | \exists z, (x,z) \in R$ ^ $(z,y) \in S\}$

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$$R^2R^3=(R R)(R R R)=R R R R R=R^5$$