Let be $B\in\mathbb R^{m \times n}$, $C\in\mathbb R^{k \times n}$, $D\in\mathbb R^{l \times n}$, $x\in\mathbb R^{n}$, $y\in\mathbb R^{m}$, $z\in\mathbb R^{k}$ and $u\in\mathbb R^{l}$, then
- $\exists x: Bx\le0, Bx\ne0, Cx\le0, Dx=0$
- $\nexists(y,z,u): B^Ty+C^Tz+D^Tu=0, z\gt 0, u\ge0$
are equivalent.
I suppose the statements should somehow be converted to other for and then use Farkas' lemma or Goldman's theorem.
