Here is a proof that uses directly the [separating hyperplane theorem][1], which states the following.

Given a matrix $A$ and column matrix $b$,
\begin{align}
Ax=b, &\text{ for some column matrix }x\ge0 \\
&\iff \\
y^TA\ge0 &\text{ for some column matrix }y \implies y^Tb\ge0
\end{align}

We will put the required proposition into the above form.
$$\begin{bmatrix}
A & b \\
0 & 1
\end{bmatrix}\begin{bmatrix}
-xx_1 \\
x_1
\end{bmatrix}\ge 0
\iff
Ax\le b \implies a'^Tx\le b' \iff \begin{bmatrix}a'^T &b'\end{bmatrix} \begin{bmatrix}
-xx_1 \\
x_1
\end{bmatrix}\ge0
 $$


  [1]: https://en.wikipedia.org/wiki/Hyperplane_separation_theorem