The author proves the following theorem: Let \(G\) be a Lie group with only finitely many components. Furthermore, let \(G\) act real-analytically and properly on the real-analytical manifolds \(X\) and \(Y\). If there is a smooth \(G\)-equivariant diffeomorphism from \(X\) to \(Y\) then there is a real-analytic \(G\)-equivariant diffeomorphism from \(X\) to \(Y\). The proof involves the existence of a global slice for the action of \(G\) and the use of a nonlinear averaging process called the center of mass construction.