The lecture note Here at last paragraph of page 136 explained the concept of 'repeated information projection' without much proof. I am wondering how can I prove the following property. Any help would be appreciated
Repeated information projection
If $Q_1$ is the information projection of $P$ onto $\Lambda_1$ (that is,
$$ D\left(Q_1 \parallel P \right) = \inf_{Q \in \Lambda_1}D\left(Q \parallel P\right). $$
Also, the $Q_2$ is the information projection of $Q_1$ onto $\Lambda_1 \cap \Lambda_2$ (that is, $$ D\left(Q_2 \parallel Q_1 \right) = \inf_{R \in \Lambda_1 \cap \Lambda_2}D\left(R \parallel Q_1\right). $$
Then $D\left(Q_2 \parallel P \right) = \inf_{S \in \Lambda_1 \cap \Lambda_2} D\left(S \parallel P \right)$, too.