Let $P = (\Omega, \mathcal{F}, \mathbb{P},\{ \mathcal{G}_j \}_{j \in J})$ be a D1726: Filtered probability space such that

(i)	$X : J \to \mathsf{Random}(\Omega \to \mathbb{R}^D)$ is a D5710: Euclidean real martingale on $P$

If $J$ is empty or a singleton, then the claim is vacuously true, so assume $J$ has at least two elements and let $i, j \in J$ such that $i \neq j$. Since $J$ is an ordered set and since $i$ and $j$ are distinct, we may assume without loss of generality that $i < j$. Since $X$ is a martingale on $P$, then $\mathbb{E}(X_j \mid \mathcal{G}_i) \overset{a.s.}{=} X_i$ and thus \begin{equation} \mathbb{E}(\mathbb{E}(X_j \mid \mathcal{G}_i)) = \mathbb{E} (X_i) \end{equation} On the other hand, result R2150: Expectation of conditional expectation for a random euclidean real number shows that \begin{equation} \mathbb{E}(\mathbb{E}(X_j \mid \mathcal{G}_i)) = \mathbb{E}(X_j) \end{equation} Combining the above two equations then gives \begin{equation} \mathbb{E} (X_i) = \mathbb{E}(\mathbb{E}(X_j \mid \mathcal{G}_i)) = \mathbb{E}(X_j) \end{equation} This finishes the proof. $\square$