  Menger's Edge Theorem asserts that there exist $k$ pairwise edge-disjoint
paths between two vertices in an undirected graph if and only if a deletion of
any $k-1$ or less edges does not disconnect these two vertices. Alternatively,
there exist $k$ pairwise summand-disjoint formal sums of edges with
coefficients in $\mathbb{F}_2$, each one of which is mapped by the boundary map
to the sum of vertices $A$ and $B$, if and only if after a deletion of any
$k-1$ or less edges there still exist a formal sum of edges with coefficients
in $\mathbb{F}_2$ which is mapped by the boundary map to $A+B$.
  We extend this result to four vertices $A,B,C,D$. We prove that in an
undirected graph, in which all the vertices different from $A,B,C,D$ have even
degrees, the following two statements are equivalent: There exist $k$ pairwise
summand-disjoint formal sums of edges with coefficients in $\mathbb{F}_2$, each
one of which is mapped by the boundary map to $A+B+C+D$; After a deletion of
any $k-1$ or less edges there still exists a formal sum of edges with
coefficients in $\mathbb{F}_2$ which is mapped by the boundary map to
$A+B+C+D$.
  Equivalently, if after a deletion of any $k-1$ or less edges, the four
vertices $A,B,C,D$ can be split into two pairs of vertices, and the two
vertices in each pair then can be connected by a path so that these two paths
are edge-disjoint, then the four vertices $A,B,C,D$ can be split $k$ times into
two pairs of vertices and the two vertices in each one of these $2k$ pairs can
then be connected by a path in such a way that all these $2k$ paths are
pairwise edge-disjoint.
