## Normal vector of a half-space containing intersections of half-spaces

Hi all,
does anyone know if there exists a result that proves/disproves the following?:

"If a half-space $\{x \in R^{n}|Ax≥0 \}$ for a row vector A contains an finite intersection of half-spaces, $\cap^{m}_{i=1} \{x \in R^{n}|A_{i}x \geq 0 \}$, for some row vectors $A_{i}, i=\{1,...,m\}$, then there exists a subset K of {1,...,m} such that $A= \sum_{j \in K} v_{j}A_{j}$ with $v_{j}>0$ for all $j \in K$"

thank you.
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