S is the polyhedral set
$ S = \{ \mathbf{x} \in \mathbb{R}^{n} ; \mathbf{Ax}=\mathbf{b}, \mathbf{x} \ge \mathbf{0} \} $
and
$ H : \mathbf{c}^{T}\mathbf{x} = \beta $
with
$ \min_S ( \mathbf{c}^{T}\mathbf{x} )= \beta $
My textbook states that given the above, the set $S \cap H$ contains no line since $\mathbf{x} \ge \mathbf{0}$ in $S$. But I don't understand why.