Let A and B represent two linear inequalities:

$A:{a}_{1}{x}_{1}+...+{a}_{n}{x}_{n}\beta \x89\u20af{k}_{1}$

$B:{b}_{1}{x}_{1}+...+{b}_{n}{x}_{n}\beta \x89\u20af{k}_{2}$

If A and B is unsatisfiable (does not have solution), does the following hold in general (the conjunction of two inequalities implies the summation of them )? If so, I am looking for a formal proof?

$A\beta \x88\S B\phantom{\rule{thickmathspace}{0ex}}\beta \x9f\u0389\phantom{\rule{thickmathspace}{0ex}}A+B$

${\pi \x9d\x91\x8e}_{1}{\pi \x9d\x91\u20af}_{1}+...+{\pi \x9d\x91\x8e}_{n}{\pi \x9d\x91\u20af}_{n}\beta \x89\u20af\pi \x9d\x91\x981\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\beta \x88\S \phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}{\pi \x9d\x91\x8f}_{1}{\pi \x9d\x91\u20af}_{1}+...{\pi \x9d\x91\x8f}_{n}{\pi \x9d\x91\u20af}_{n}\beta \x89\u20af{\pi \x9d\x91\x98}_{2}\phantom{\rule{thickmathspace}{0ex}}\beta \x9f\u0389\phantom{\rule{thickmathspace}{0ex}}{\pi \x9d\x91\x8e}_{1}{\pi \x9d\x91\u20af}_{1}+...+{\pi \x9d\x91\x8e}_{n}{\pi \x9d\x91\u20af}_{n}+{\pi \x9d\x91\x8f}_{1}{\pi \x9d\x91\u20af}_{1}+...{\pi \x9d\x91\x8f}_{n}{\pi \x9d\x91\u20af}_{n}\beta \x89\u20af{\pi \x9d\x91\x98}_{1}+{\pi \x9d\x91\x98}_{2}$

and then I would like to generalize the above theorem to summation of several inequalities.

$A:{a}_{1}{x}_{1}+...+{a}_{n}{x}_{n}\beta \x89\u20af{k}_{1}$

$B:{b}_{1}{x}_{1}+...+{b}_{n}{x}_{n}\beta \x89\u20af{k}_{2}$

If A and B is unsatisfiable (does not have solution), does the following hold in general (the conjunction of two inequalities implies the summation of them )? If so, I am looking for a formal proof?

$A\beta \x88\S B\phantom{\rule{thickmathspace}{0ex}}\beta \x9f\u0389\phantom{\rule{thickmathspace}{0ex}}A+B$

${\pi \x9d\x91\x8e}_{1}{\pi \x9d\x91\u20af}_{1}+...+{\pi \x9d\x91\x8e}_{n}{\pi \x9d\x91\u20af}_{n}\beta \x89\u20af\pi \x9d\x91\x981\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\beta \x88\S \phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}{\pi \x9d\x91\x8f}_{1}{\pi \x9d\x91\u20af}_{1}+...{\pi \x9d\x91\x8f}_{n}{\pi \x9d\x91\u20af}_{n}\beta \x89\u20af{\pi \x9d\x91\x98}_{2}\phantom{\rule{thickmathspace}{0ex}}\beta \x9f\u0389\phantom{\rule{thickmathspace}{0ex}}{\pi \x9d\x91\x8e}_{1}{\pi \x9d\x91\u20af}_{1}+...+{\pi \x9d\x91\x8e}_{n}{\pi \x9d\x91\u20af}_{n}+{\pi \x9d\x91\x8f}_{1}{\pi \x9d\x91\u20af}_{1}+...{\pi \x9d\x91\x8f}_{n}{\pi \x9d\x91\u20af}_{n}\beta \x89\u20af{\pi \x9d\x91\x98}_{1}+{\pi \x9d\x91\x98}_{2}$

and then I would like to generalize the above theorem to summation of several inequalities.