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

hwangii

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

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

thank you.