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Prove that for any four positive real numbers `a, b, c, d` the inequality holds. Determine all cases of equality.
`[(a - b)(a - c)]/(a + b + c) + [(b - c)(b - d)]/(b + c + d) + [(c - d)(c - a)]/(c + d + a) + [(d - a)(d - b)]/(d + a + b) >= 0`