= Solution
{c}
Fix $A\in\mathcal A_1$ and define
$$
\mathcal D_A=\{B\in\mathcal G_2:\mathbb P(A\cap B)=\mathbb P(A)\mathbb P(B)\}.
$$
This is a <Dynkin system>: it contains $\Omega$; taking <complements> preserves the equality because $\mathbb P(A\cap B^c)=\mathbb P(A)-\mathbb P(A\cap B)$; and <countable additivity> preserves it under disjoint <unions>. The hypothesis says that $\mathcal D_A$ contains the <pi-system> $\mathcal A_2$. Hence <Dynkin lemma> gives $\mathcal D_A=\mathcal G_2$.
Now fix any $B\in\mathcal G_2$ and put
$$
\mathcal E_B=\{A\in\mathcal G_1:\mathbb P(A\cap B)=\mathbb P(A)\mathbb P(B)\}.
$$
The same <complement> and disjoint-<union> calculations make this a <Dynkin system>. The first step shows that it contains $\mathcal A_1$, so <Dynkin lemma> gives $\mathcal E_B=\mathcal G_1$. Thus
$$
\boxed{\mathbb P(A\cap B)=\mathbb P(A)\mathbb P(B)\quad(A\in\mathcal G_1,\ B\in\mathcal G_2).}
$$
This is precisely the definition of <independent sigma-algebras>, and proves <independence extended from generating pi-systems>.
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