Real Number Inequalities can be Added

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Theorem

Let $a, b, c, d \in \R$ such that $a > b$ and $c > d$.

Then $a + c > b + d$.

	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle a$	$>$	$\displaystyle b$	$\displaystyle $	$\displaystyle $	$\displaystyle $
	$\displaystyle $	$\displaystyle \implies$	$\displaystyle $	$\displaystyle a + c$	$>$	$\displaystyle b + c$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Real Number Ordering is Compatible with Addition

	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle c$	$>$	$\displaystyle d$	$\displaystyle $	$\displaystyle $	$\displaystyle $
	$\displaystyle $	$\displaystyle \implies$	$\displaystyle $	$\displaystyle b + c$	$>$	$\displaystyle b + d$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Real Number Ordering is Compatible with Addition

Finally:

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle a + c$	$>$	$\displaystyle b + c$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle b + c$	$>$	$\displaystyle b + d$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle \implies$	$\displaystyle $	$\displaystyle a + c$	$>$	$\displaystyle b + d$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Trichotomy Law for Real Numbers

$\blacksquare$

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle a\)	\(>\)	\(\displaystyle b\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle a + c\)	\(>\)	\(\displaystyle b + c\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Real Number Ordering is Compatible with Addition

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle c\)	\(>\)	\(\displaystyle d\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle b + c\)	\(>\)	\(\displaystyle b + d\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Real Number Ordering is Compatible with Addition

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle a + c\)	\(>\)	\(\displaystyle b + c\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle b + c\)	\(>\)	\(\displaystyle b + d\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle a + c\)	\(>\)	\(\displaystyle b + d\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Trichotomy Law for Real Numbers