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In this paper, we first study the relations between semiprime ideals and ideals of the ideal lattice of posets. Moreover, we study the relation between prime ideals of a poset P and prime ideals of the set Id(P) of all ideals of P. We extend separation theorem for semiprime to a finite up directed poset with some constraint on the filter and for prime ideals to the case of finite poset. Then we prove the prime separation theorem for finite posets P for which Id(P) is distributive.