# Proof that $E(X|\sigma (G_1,G_2))=E(X|G_1)$, when $G_2,\sigma(\sigma(X),G_1)$ independent.

I want to prove:

$$E(X|\sigma (G_1,G_2))=E(X|G_1)$$

when $$G_2,\sigma(\sigma(X),G_1)$$ are independent. It's quite elusive expression to unpack, what would you say is the meaning? Previously I have proven that:

$$L=\{A\in F| E(I_AX)=E(I_AE(X|G_1)\}$$ is a $$\lambda$$-system and contains the set: $$\{ A\cap B| A\in G_1,B\in G_2\}$$. Note that $$F$$ is the simga algebra of the space which $$G_1,G_2$$ are subsets of.

How to continue from here?