Let X and Y be locally compact Hausdorff spaces. Let A and B be complex-linear subspaces of C-0(X) and C-0(Y), respectively. Suppose that for each triple of distinct points x, x', x" epsilon X, there exists f epsilon A such that vertical bar f(x)vertical bar not equal vertical bar f(x')vertical bar and f (x") = 0. Also suppose that for each pair of distinct points y, y' epsilon Y. there exists g epsilon B such that vertical bar g(y)vertical bar not equal vertical bar g(y')vertical bar. For such A and B, we prove the following statement: If T is a real-linear isometry of A onto B, then there exist an open and closed subset E of ChB, a homeomorphism phi of ChB onto Ch A and a unimodular continuous function omega on ChB such that Tf = omega(f o phi) on E and Tf = omega(f o phi) on ChB\E for all f epsilon A, where Ch A and ChB are the Choquet boundaries for A and B, respectively. Moreover, we remark that the separation condition on A cannot be omitted in the above result. (C) 2013 Elsevier Inc. All right's reserved.