= Matroid closure exchange property
{title2=$y\notin\operatorname{cl}_M(A),\ y\in\operatorname{cl}_M(A+x)\Rightarrow x\in\operatorname{cl}_M(A+y)$}
The <closure of a matroid> obeys the displayed exchange rule. This is the finite <pregeometry> exchange principle: if adding $x$ newly makes $y$ dependent, then adding $y$ also makes $x$ dependent.
\b[Proof.] Write $r=r(A)$. The first hypothesis and the one-element bound on the <rank of a matroid> give $r(A+y)=r+1$. The second gives $r(A+x+y)=r(A+x)\leq r+1$. Inclusion gives $r(A+x+y)\geq r(A+y)$, so equality holds. This is exactly $x\in\operatorname{cl}_M(A+y)$.
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