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 :

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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 ?