Proof of addition and subtraction rules for systems of inequalities. I've heard it claimed that for a system of inequalities such as: 
$a<b$ 
and $ c<d $ 
Then $a+c<b+d$ is a valid move, but subtraction isn't.
And similarly for a system such as $e>f$ and $g<h$ then $e-g>f-h$ is valid and so is $g-e<h-f$, but addition isn't valid.
In short, I've been told that you can only add inequalities when they both have the same inequality sign (I believe the term is "sense") and you can only subtract them when they don't.
However, I have neither seen nor can I derive, a prove that these rules are true.
Does anyone have such a proof?

Given $a\lt b$ and $c\lt d$ then $a+c\lt b+c\lt b+d$
Given $e\gt f$ and $g\lt h$ then $e-g\gt f-g\gt f-h$
