How to prove a relation of conditional expectation?

Question

If $\mathcal G,\mathcal H$ are two fields with $\mathcal G \supset \mathcal H$ and if $\mathbb E(Z|\mathcal G),\mathbb E(Z|\mathcal H)$ have the same distribution, then why we can prove $\mathbb E(\mathbb E(Z|\mathcal G)\mathbb E(Z|\mathcal H))={\mathbb E(\mathbb E(Z|\mathcal H))}^{2}$?

Answer 1

Since $H \subset G$, we have that $$ E(E(Z \vert G) \vert H) = E(Z \vert H). $$ So $$ E(E(Z \vert G) E(Z \vert H) ) = E(E(E(Z \vert G) \vert H) E(Z \vert H) ) = E( [ E(Z \vert H)]^2 ) $$