Let $(Y,\mathcal{C},\lambda,T)$ be a Bernoulli system measure-theoretically isomorphic to a two-sided Bernoulli shift over a one-coordinate probability space $(A,\mathcal{A},\nu)$, and assume that the one-coordinate partition has at least two atoms of positive measure. Then $(Y,\mathcal{C},\lambda,T)$ has the K-property.