  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.
