Boundedly connected sets and the distance to the intersection of two sets

We prove that if (X,d) is a metric space, C is a closed subset of X and x∈X, then the distance of x to R∩S agrees with the maximum of the distances of x to R and S, for every closed subsets R,S⊂C such that C=R∪S, if and only if C is x-boundedly connected.