Can anyone give me an explanation why $E(Y| X) = E(E(Y| X, Z)| X)$? I am a rookie in probability. Can anyone give me an intuitive explanation and a rigorous mathematical proof to why $E(Y| X) = E(E(Y| X, Z)| X)$? Thanks in advance!

Intuitive explanation, which happens to be fully rigorous when $Y$ is square integrable: let $U\subseteq V$ denote some vector subspaces of an inner product space $W$, then the orthogonal projections $\mathrm{pr}_U:W\to W$ on $U$ and $\mathrm{pr}_V:W\to W$ on $V$ are such that $$ \mathrm{pr}_U\circ\mathrm{pr}_V=\mathrm{pr}_U.$$
If $U=L^2(\sigma(X))$, $V=L^2(\sigma(X,Z))$ and $W=L^2$, then $\mathrm{pr}_U=E(\ \mid X)$ and $\mathrm{pr}_V=E(\ \mid X,Z)$.
