On image-weight and image-distance inequalities

Jang and Park asked in [On a MacWilliams type identity and a perfectness for a binary linear image-poset code, Discrete Math. 265 (2003) 85–104] whether, for each poset image, the P-weights and P-distances satisfy the inequalities image for all vectors image. We prove that these inequalities hold for all vectors image over any field image.