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By taking into account the work of {\it J. Rataj} and {\it M. Z\"ahle} [Geom. Dedicata 57, 259-283 (1995; Zbl 0844.53050)], {\it R. Schneider} and {\it W. Weil} [Math. Nachr. 129, 67-80 (1986; Zbl 0602.52003)], {\it W. Weil} [Math. Z. 205, 531-549 (1990; Zbl 0705.52006)], an integral formula is obtained here by using the technique of rectifiable currents.\par This is an iterated version of the principal kinematic formula for $q$ sets of positive reach and generalized curvature measures.
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