In this paper the following theorem is obtained: \emph{Let $X$ be a Hausdorff locally convex space, $\{p_i\}i\in I$ be a saturated family of seminorms defining the topology of $X$, $f$ be a mapping of $I$ into $I$ and $F$ be a mapping of $X$ into $X$ satisfying the following conditions}: \emph{For every $i\in I$ there exists $q(i)>0$ and $f:I\to I$ such that}: \begin{enumerate} ıem $p_i(Fx-Fy)\leq q(i)p_{f(i)}(x-y)$ \emph{for every} $x$, $y\in X$; ıem $q(f^k(i))\leq Q(i)<1$, $k=1,2,\ldots$ \emph{for every} $i\in I$; ıem $(f^{k(i)}(i))=f(i)$ \emph{for every} $i\in I$, $k(i)\in N$ Then $F\in D(X)$, \emph{whwere} $D=\{p_i\}i\in I$. \end{enumerate}