  We show that for a convex function the following, rather modest conditions,
are equivalent to monotonicity under local operations and classical
communication. The conditions are: 1)invariance under local unitaries, 2)
invariance under adding local ancilla in arbitrary state 3) on mixtures of
states possessing local orthogonal flags the function is equal to its average.
The result holds for multipartite systems. It is intriguing that the obtained
conditions are equalities. The only inequality is hidden in the condition of
convexity.
