add a proof

main.tex

@@ -182,6 +182,13 @@ The size of every $k'$-cocycle is $k'+n$, and for any $k'$-cocycle there are bas
 Let $M=(E,\mathcal I)$ be a matroids and let $F$ be a flat of $M$.
 Then $\cl(F+e)\setminus F$ is a partition of $E\setminus F$.
 \end{proposition}
+\begin{proof}
+Suppose for contradiction that there are two elements $x,y\in E-F$ such that $\cl(F+x)\setminus F$ and $\cl(F+y)\setminus F$ have non-empty intersection. Let $z$ be an element in the intersection.
+Then we find a circuit $C_{x,z}$ in $\cl(F+e)$ such that $C_{x,z}\setminus F= \set{x,z}$.
+Note that this circuit exists since $z$ is in the span of $F+e$.
+Let $C_{y,z}$ denote the analogous circuit for $y$.
+Then it follows from the circuit axiom that there is another circuit $C\subset C_{x,z}\cup C_{y,z}\setminus \set{z}$, which implies $y\in \cl(F+x)$ and thus contradicts the assumption.
+\end{proof}
 
 \subsection{Support size}