As of now, I know the concept of projection that the orthogonal projection of a set $S ⊂ \mathbb{R}^{n+p}$ onto the linear subspace $\mathbb{R}^n × \{0\}^p$ is $$proj_x(S) := {x ∈ \mathbb{R}^n : ∃z ∈ \mathbb{R}^p \quad s.t. (x, z) ∈ S}$$
But I have problems with how to apply it practically with several doubtful tries. Any help with this issue considering the following example is much appreciated.
Consider a set S described by the following system
$$ x_1 = z_1 + w_1, \\ x_2 = z_1 + w_2, \\ w_1 ≤ z_1, \\ w_2 ≤ z_1, \\ w_1 ≥ 0, \\ w_2 ≥ 0, \\ z_1 ∈ \mathbb{Z}$$
How to plot the feasible region of S projected down to the space of variables $x_1$ and $x_2$ and find its corners points within a limited range of $x_1$ and $x_2$.
You can infer from $0\le w_1 \le z_1$ that $z_1\ge 0$. Now eliminate $w$ to obtain the inequalities $z_1 \le x_i \le 2z_1$ ($i=1,2$) and you can easily plot the feasible values of $x$.