Properties of Two Inequalities If $a &lt; b$ and $c &lt; d$, then $a + c &lt; b + d$ if $a, b, c, d$ are positive integers.

My hunch is that this is true, but I'm having difficulty proving it.  Is there a property or definition pertaining to inequalities that exists to support/disprove the above?
$a &lt; b \implies a+c &lt; b+c$

$c &lt; d \implies b+c &lt; b+d$

So $a &lt; b$ and $c&lt;d \implies a+c &lt; b+c &lt; b+d \implies a+c &lt; b+d$
$a&lt;b \implies a+c &lt; b+c &lt;b+d$