In this paper we first give a description of a surjective unit-preserving real-linear uniform isometry $ T : A \longrightarrow B$, where $ A $ and $ B $ are complex function spaces on compact Hausdorff spaces $ X $ and $ Y $, respectively, whenever ${\rm ER}\left (A, X\right ) = {\rm Ch}\left (A, X\right )$ and ${\rm ER}\left (B, Y\right ) = {\rm Ch}\left (B, Y\right )$. Next, we give a description of $ T $ whenever $ A $ and $ B $ are complex function algebras and $ T $ does not assume to be unit-preserving.
