For any non-zero natural integer $n$, we set
$$\omega(n) = \operatorname{Card}\{p \text{ prime} : p \mid n\} = \sum_{\substack{p \mid n \\ p \text{ prime}}} 1.$$
Show that
$$\sum_{n \leqslant x} \omega(n)^2 = \sum_{\substack{p_1 \leqslant x \\ p_1 \text{ prime}}} \sum_{\substack{p_2 \leqslant x \\ p_2 \text{ prime}}} \operatorname{Card}\left\{n \in \mathbb{N}^* : n \leqslant x, p_1 \mid n \text{ and } p_2 \mid n\right\}$$