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W. Haebich (1972, $Bull. Austral. Math. Soc.$ \textbf{7},279-296)
presented a formula for the Schur multiplier of a regular product
of groups. In this paper, first, it is shown that the
Baer-invariant of a nilpotent product of groups with respect to
the variety of nilpotent groups has a homomorphic image and in
finite case a subgroup of Haebich's type. Second, a formula will
be presented for the Baer-invariant of a nilpotent product of
cyclic groups with respect to the variety of nilpotent groups.
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