Let $Z=\{(x,y)\in R^d\times R^l: a\leq Ax\leq b, y=Bx\}$ where $A,B$ are matrices and $a, b$ are vectors in $R^m$, with $a_i\leq b_i$, for all $i=1,...,m$.
Then $Z=Z_1\cap Z_2$ where
$Z_1=\{(x,y): Ax\leq b, y=Bx\}$,
$Z_2=\{(x,y): Ax\geq a, y=Bx\}$.
For any $P\subset R^d\times R^l$, define the projection of $P$ onto $y$-space by
$proj_y(P)=\{y\in R^l: \exists x\in R^d \,\,\,such\,\, that\,\,\, (x,y)\in P\}.$
I want to show that the projection of the intersection of the projections of $Z_1$ and $Z_2$ onto $y$-space equals to the projection of $Z$ onto $y$-space, i.e.
$$Proj_y (Z_1)\cap Proj_y (Z_2)=Proj_y (Z).$$
Thanks in advance.