  We consider a compact $C^\omega$ manifold $X$ and finitely many regular
$C^\omega$ submanifolds $Y_1, \dots, Y_q$ of $X$, which are closed subsets in
$X$, such that the union of $Y_j$'s has only normal crossings. We show that
every continuous function on the union which is of class $C^\omega$ on each
$Y_j$ can be extended to a $C^\omega$ function on $X$. A crucial feature of our
proof is to employ basic tools of real analytic geometry -- Cartan Theorems A
and B.
