real analysis - Solving Basel Problem using euler infinite product and infinite product-sum equality

Trying to prove Basel problem through the equality $sin(x) = x\prod\limits_{k=1}^{+\infty}(1-\frac{x^{2}}{\pi^{2}k^{2}})$,

I came across the following problem;

I was able to prove the following equality by induction in a finite case, I'd like to prove the general one,which is,for $\{a_{k}\} \subseteq \mathbb{R}$ :

$$\prod_{k \in I}(1+a_{k}) = \sum\limits_{n \in \mathbb{N}}(\sum\limits_{\underset{J \in F(\mathbb{N_{+}})}{\lvert J \rvert = n}} \prod_{k \in J}a_{k})$$

Where $F(\mathbb{N_{+}})$ denotes the finite subsets on $\mathbb{N_{+}}$.

Any tip,suggestion or sketch of the proof would be appreciated.