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We study the Euler characteristic modulo 2 of the manifold immersed in the multiple point set of the self-transverse immersions of the real projective space RP n and the Milnor manifold H m,n in the sphere. Mathematics Subject Classification: Primary 57R42; Secondary 57R20, 55N15.doi:10.2206/kyushujm.60.331 fatcat:55xob5cjxbdi5ouzn5fchovys4