If $C:=\{x:Ax\leq 0\}\neq\{x:Ax=0\}$, an independent set of rows of $A$ can be chosen, one denoted by $a$ and the others put as rows into a matrix $A'$, such that $\{x:A'x=0,ax\leq 0\}\subseteq C$. Why?
This is asserted in a proof (which I don't understand yet) that polyhedral cones (sets of type $\{x:Ax\leq 0\}$ in $\mathbb{R}^n$) are finitely generated cones (there exists a finite set of vectors such that the set of linear combinations with positive coefficients of these equals the cone).