# How to prove that the dual linear program of the max-flow linear program indeed is a min-cut linear program?

So the wikipedia page gives the following linear programs for max-flow, and the dual program :

While it is quite straight forward to see that the max-flow linear program indeed computes a maximum flow (every feasable solution is a flow, and every flow is a feasable solution), i couldn’t find convincing proof that the dual of the max-flow linear program indeed is the LP of the min-cut problem.

An ‘intuitive’ proof is given on wikipedia, namely : $$d_{uv}$$ is 1 if the edge $$(u,v)$$ is counted in the cut and else $$0$$, $$z_u$$ is $$1$$ if $$u$$ is in the same side than $$s$$ in the cut, and $$0$$ if $$u$$ is in the same side of the cut than $$t$$

But that doesn’t convince me a lot, mainly why should all the variables be integers, while we don’t have integer conditions ?

And in general, do you have a convincing proof that the dual of the max-flow LP indeed is the LP formulation for min-cut ?