A polynomial system \([P,Q]\) is a pair of multivariate polynomial sets \(P\) and \(Q\) and Zero \(([P,Q])\) is the set of all common zeros of the polynomials in \(P\) which are not zeros of any polynomial in \(Q\). Fixing an ordering to the variables, a simple system is a polynomial system ordered in triangular form, in which every polynomial is squarefree and has non-vanishing leading coefficient with respect to its leading variable. The author shows how to decompose any polynomial system into finitely many simple systems \([T_i,T_i']\) such that \[ \text{Zero} \bigl([P,Q]\bigr) =\bigcup \text{Zero} [T_i,T_i']. \] The method deals with top-down elimination and the construction of subresultant chains. The author also presents an algorithm (Sim Sys) that decomposes any polynomial system in the sense stated above. He shows how Sim Sys works in different examples and finally states several properties of simple systems.