Is the following argument Corrrect?
Problem. Two experiments are to be performed. The first can result in any one of $m$ possible outcomes. If the first experiment results in outcome $i$, then the second experiment can result in any of $n_i$ possible outcomes, $i = 1, 2, . . . ,m.$ What is the number of possible outcomes of the two experiments?
Proof. Let $I_{m} = \{1,2,\dots,m\}$ and $\forall i\in I_m$ define $E_{i}$ to be the set of of all out comes of the second experiment given that the outcome of experiment one was $i$. Furthermore define $O$ to be the set of all possible out comes of the above described process that is $O = \{(i,j)\ |\ i\in I_m\text{ and }j\in E_i\}$ $$|O| = \left|\bigcup_{i\in I_m}\{i\}\times E_{i}\right| = \sum_{i\in I_m}|E_i| = \sum_{i=1}^{m}n_i$$