Hilbert functions of monomial ideals containing a regular sequence

Let M be an ideal in K[x(1),...,x (n)] (K is a field) generated by products of linear forms and containing a homogeneous regular sequence of some length. We prove that ideals containing M satisfy the Eisenbud-Green-Harris conjecture and moreover prove that the Cohen-Macaulay property is preserved. We conclude that monomial ideals satisfy this conjecture. We obtain that the h-vector of Cohen-Macaulay simplicial complex Delta is the h-vector of Cohen-Macaulay (a (1) - 1,..., a (t) - 1)-balanced simplicial complex, where t is the height of the Stanley-Reisner ideal of Delta and (a (1),..., a (t)) is the type of some regular sequence contained in this ideal.