= Solution
Since $Y$ is a maximal <matroid independent set> in $Z$, augmentation gives $r(Y)=|Y|=r(Z)$. For any $x\in X$, the <rank of a matroid> of $Z+x$ equals that of $Y+x$. Indeed, if $x\notin Z$, either $Y+x$ is independent and has the largest possible size $|Y|+1$, or it is dependent and $Y$ is still maximal in $Z+x$: no element of $Z\setminus Y$ can be added, and neither can $x$. If $x\in Z$, both ranks are simply $|Y|$.
Consequently $r(Y+x)=r(Y)$ if and only if $r(Z+x)=r(Z)$, which proves \b[$\sigma(Y)=\sigma(Z)$] for the <closure of a matroid>.
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