A Stronger Multiple Exchange Property for M^(natural)-concave Functions

The multiple exchange property for matroid bases has recently been generalized for valuated matroids and M$^{\natural}$-concave set functions. This paper establishes a stronger form of this multiple exchange property that imposes a cardinality condition on the exchangeable subset. The stronger form immediately implies the defining exchange property of M$^{\natural}$-concave set functions, which was not the case with the recently established multiple exchange property without the cardinality condition.