Then $p$ is a valuated matroid, meaning that $\mathcal{B}\neq\varnothing$ and for every $S\in\binom{E}{r-1}$ and every $T\in\binom{E}{r+1}$ the minimum
is attained at least twice, if and only if $\mathcal{B}\neq\varnothing$ and for every $B,B'\in\mathcal{B}$ and every $e\in B\setminus B'$, there exists $f\in B'\setminus B$ such that