Ordering is Preserved on Integers by Addition

Theorem
The usual ordering relation on the integers is preserved by the operation of addition:


 * $\forall a, b, c, d, \in \Z: a \le b, c \le d \implies a + c \le b + d$

Proof
Recall that Integers form Ordered Integral Domain.

Then from Relation Induced by Positivity Property is Compatible with Addition:


 * $\forall x, y, z \in \Z: x \le y \implies \left({x + z}\right) \le \left({y + z}\right)$
 * $\forall x, y, z \in \Z: x \le y \implies \left({z + x}\right) \le \left({z + y}\right)$

So:

Finally: