Independence of sigma-algebras is hereditary

Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
 (i) $\mathcal{G}_j \subseteq \mathcal{F}$ is a D470: Subsigma-algebra of $\mathcal{F}$ on $\Omega$ for each $j \in J$ (ii) $\mathcal{H}_j \subseteq \mathcal{G}$ is a D470: Subsigma-algebra of $\mathcal{G}_j$ on $\Omega$ for each $j \in J$ (iii) $\mathcal{G} = \{ \mathcal{G}_j \}_{j \in J}$ is an D471: Independent collection of sigma-algebras on $P$
Then $\mathcal{H} = \{ \mathcal{H}_j \}_{j \in J}$ is an D471: Independent collection of sigma-algebras on $P$.
Proofs

Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
 (i) $\mathcal{G}_j \subseteq \mathcal{F}$ is a D470: Subsigma-algebra of $\mathcal{F}$ on $\Omega$ for each $j \in J$ (ii) $\mathcal{H}_j \subseteq \mathcal{G}$ is a D470: Subsigma-algebra of $\mathcal{G}_j$ on $\Omega$ for each $j \in J$ (iii) $\mathcal{G} = \{ \mathcal{G}_j \}_{j \in J}$ is an D471: Independent collection of sigma-algebras on $P$
If $E_{j_1} \in \mathcal{H}_{j_1}, \ldots, E_{j_N} \in \mathcal{H}_{j_N}$, then $E_{j_1} \in \mathcal{G}_{j_1}, \ldots, E_{j_N} \in \mathcal{G}_{j_N}$ which implies that $E_{j_1}, \ldots, E_{j_N}$ are independent events in $P$ since $\mathcal{G}_{j_1}, \ldots, \mathcal{G}_{j_N}$ are independent sigma-algebras. $\square$