Let $N\in\mathbb N$, let $M\subset\mathbb C^N$ be a real-analytic generic CR submanifold, and let $f:M\to\mathbb C$ be a real-analytic CR function. For $p\in M$, let $\mathcal O(p)$ denote the CR orbit of $p$, equipped with its immersed real-analytic manifold topology. If $C$ is a [connected component](/page/Connected%20Component) of $\mathcal O(p)$ and if there exists a non-empty open subset $A\subset C$ such that $f|_A=0$, then $f|_C=0$.