  We study the composition of an arbitrary number of Fourier integral operators
$A_j$, $j=1,\dots,M$, $M\ge 2$, defined through symbols belonging to the
so-called SG classes. We give conditions ensuring that the composition
$A_1\circ\cdots\circ A_M$ of such operators still belongs to the same class.
Through this, we are then able to show well-posedness in weighted Sobolev
spaces for first order hyperbolic systems of partial differential equations
with coefficients in SG classes, by constructing the associated fundamental
solutions.
