Suppose that $A$ is an $m\times n$ matrix, $D$ is a $p\times n$ matrix, $b$ is an $m$-vector, and $d$ is a $p$-vector. Prove that there does not exist $n$-vector $x$ satisfying
$Ax \geq b, Dx \leq d$
if and only if there exist $m$-vector $y$ and $p$-vector $w$ satisfying
$ y \le 0, w \ge 0,$
$ (A^T)y + (D^T)w = 0,$
$ (b^T)y + (d^T)w < 0.$