Subsets and Sigma Algebras: Proving the Equality of Generated Sigma Algebras 

Let $X$ a set and $Y$ a subset of $X$. Let $B$ a sigma algebra over $X$ and $F=\left\{ E\cap Y:\,E\in B\right\}$
a sigma algebra over $Y$.

If $B=\sigma(\mathcal{E}_{1})$ and $\mathcal{E}_{2}=\left\{ E\cap Y: \,E\in \mathcal{E}_{1}\right\}$, prove that $F=\sigma(\mathcal{E}_{2}).$

Prove that 

I) $F\subset \sigma(\mathcal{E}_{2}) $
II) $\sigma(\mathcal{E}_{2}) \subset F.$

The answer is accepted.
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