ISL 1970 Problem 26

Consider a finite set of vectors in space $\{a_1, a_2, ... , a_n\}$ and the set $E$ of all vectors of the form $x=\sum_{i=1}^{n}{\lambda _i a_i}$ , where $\lambda _i \in \mathbb{R}^{+}\cup \{0\}$ . Let $F$ be the set consisting of all the vectors in $E$ and vectors parallel to a given plane $P$ . Prove that there exists a set of vectors $\{b_1, b_2, ... , b_p\}$ such that $F$ is the set of all vectors $y$ of the form $y=\sum_{i=1}^{p}{\mu _i b_i}$ , where $\mu _i \in \mathbb{R}^{+}\cup \{0\}$ .