Suppose $G=(V,E)$ be a directed graph , and let $u,v,w$ be distinct vertices. Suppose there are $k$ edge disjoint paths from $u$ to $v$ in $G$, and $k$ edge disjoint paths from $v$ to $w$ in $G$. The paths from $u$ to $v$ van share edges with the paths from $v$ to $w$. Then, how to show that there are $k$ edge disjoint paths from $u$ to $w$ in $G$ using Menger's theorem or MaxFlow-MinCut.
Update: I have already proved the following statement: Given an integer $k>0$, $G$ has $k$ edge disjoint paths from $s$ to $t$ if and only if there is an $s,t$ flow of value $k$ in $G$.
By the above statement, we know that $G$ has a $u,v$ flow of value k, and $G$ has a $v,w$ flow of value k, but how can we show there is a $u,w$ flow of value $k$ in $G$? This means I have to prove the transitivity of a flow, my thought is to use the flow conservation property of internal vertex to prove that, the internal vertex which joins the $u,v$ flow and $v,w$ flow is $v$. Is this the right approach or there is a better way to approach this?
An $x,y$-flow of value $k$ is a flow with excess $-k$ at $x$ ($k$ more flow leaving than entering), excess $+k$ at $y$ ($k$ more flow entering than leaving) and excess $0$ at every other node (flow conservation).
So if you add together a $u,v$-flow of value $k$ and a $v,w$-flow of value $k$ (edge by edge), you get something that's almost like a $u,w$-flow of value $k$. The only problem is that some edges might be used twice in the sum, exceeding capacity.
Prove that if an edge $xy$ is used twice in the sum of the two flows, then the sum of the flows also contains a cycle containing $xy$. Then, we can subtract $1$ from the flow along each edge of the cycle, and avoid this problem. Repeat for every such edge, and you'll get an actual $u,w$-flow of value $k$.